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Theorem fin1a2lem13 9178
Description: Lemma for fin1a2 9181. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
Assertion
Ref Expression
fin1a2lem13 (((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) → ¬ (𝐵𝐶) ∈ FinII)

Proof of Theorem fin1a2lem13
Dummy variables 𝑒 𝑓 𝑔 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 477 . . 3 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) → (𝐵𝐶) ∈ FinII)
2 simpll1 1098 . . . 4 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) → 𝐴 ⊆ 𝒫 𝐵)
3 ssel2 3578 . . . . . . . . . 10 ((𝐴 ⊆ 𝒫 𝐵𝑔𝐴) → 𝑔 ∈ 𝒫 𝐵)
43elpwid 4141 . . . . . . . . 9 ((𝐴 ⊆ 𝒫 𝐵𝑔𝐴) → 𝑔𝐵)
54ssdifd 3724 . . . . . . . 8 ((𝐴 ⊆ 𝒫 𝐵𝑔𝐴) → (𝑔𝐶) ⊆ (𝐵𝐶))
6 sseq1 3605 . . . . . . . 8 (𝑓 = (𝑔𝐶) → (𝑓 ⊆ (𝐵𝐶) ↔ (𝑔𝐶) ⊆ (𝐵𝐶)))
75, 6syl5ibrcom 237 . . . . . . 7 ((𝐴 ⊆ 𝒫 𝐵𝑔𝐴) → (𝑓 = (𝑔𝐶) → 𝑓 ⊆ (𝐵𝐶)))
87rexlimdva 3024 . . . . . 6 (𝐴 ⊆ 𝒫 𝐵 → (∃𝑔𝐴 𝑓 = (𝑔𝐶) → 𝑓 ⊆ (𝐵𝐶)))
9 vex 3189 . . . . . . 7 𝑓 ∈ V
10 eqid 2621 . . . . . . . 8 (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑔𝐴 ↦ (𝑔𝐶))
1110elrnmpt 5332 . . . . . . 7 (𝑓 ∈ V → (𝑓 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) ↔ ∃𝑔𝐴 𝑓 = (𝑔𝐶)))
129, 11ax-mp 5 . . . . . 6 (𝑓 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) ↔ ∃𝑔𝐴 𝑓 = (𝑔𝐶))
13 selpw 4137 . . . . . 6 (𝑓 ∈ 𝒫 (𝐵𝐶) ↔ 𝑓 ⊆ (𝐵𝐶))
148, 12, 133imtr4g 285 . . . . 5 (𝐴 ⊆ 𝒫 𝐵 → (𝑓 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) → 𝑓 ∈ 𝒫 (𝐵𝐶)))
1514ssrdv 3589 . . . 4 (𝐴 ⊆ 𝒫 𝐵 → ran (𝑔𝐴 ↦ (𝑔𝐶)) ⊆ 𝒫 (𝐵𝐶))
162, 15syl 17 . . 3 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) → ran (𝑔𝐴 ↦ (𝑔𝐶)) ⊆ 𝒫 (𝐵𝐶))
17 simplrr 800 . . . 4 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) → 𝐶𝐴)
18 difid 3922 . . . . . . 7 (𝐶𝐶) = ∅
1918eqcomi 2630 . . . . . 6 ∅ = (𝐶𝐶)
20 difeq1 3699 . . . . . . . 8 (𝑔 = 𝐶 → (𝑔𝐶) = (𝐶𝐶))
2120eqeq2d 2631 . . . . . . 7 (𝑔 = 𝐶 → (∅ = (𝑔𝐶) ↔ ∅ = (𝐶𝐶)))
2221rspcev 3295 . . . . . 6 ((𝐶𝐴 ∧ ∅ = (𝐶𝐶)) → ∃𝑔𝐴 ∅ = (𝑔𝐶))
2319, 22mpan2 706 . . . . 5 (𝐶𝐴 → ∃𝑔𝐴 ∅ = (𝑔𝐶))
24 0ex 4750 . . . . . 6 ∅ ∈ V
2510elrnmpt 5332 . . . . . 6 (∅ ∈ V → (∅ ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) ↔ ∃𝑔𝐴 ∅ = (𝑔𝐶)))
2624, 25ax-mp 5 . . . . 5 (∅ ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) ↔ ∃𝑔𝐴 ∅ = (𝑔𝐶))
2723, 26sylibr 224 . . . 4 (𝐶𝐴 → ∅ ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)))
28 ne0i 3897 . . . 4 (∅ ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) → ran (𝑔𝐴 ↦ (𝑔𝐶)) ≠ ∅)
2917, 27, 283syl 18 . . 3 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) → ran (𝑔𝐴 ↦ (𝑔𝐶)) ≠ ∅)
30 simpll2 1099 . . . 4 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) → [] Or 𝐴)
31 vex 3189 . . . . . . . 8 𝑥 ∈ V
3210elrnmpt 5332 . . . . . . . 8 (𝑥 ∈ V → (𝑥 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) ↔ ∃𝑔𝐴 𝑥 = (𝑔𝐶)))
3331, 32ax-mp 5 . . . . . . 7 (𝑥 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) ↔ ∃𝑔𝐴 𝑥 = (𝑔𝐶))
34 difeq1 3699 . . . . . . . . . 10 (𝑔 = 𝑒 → (𝑔𝐶) = (𝑒𝐶))
3534eqeq2d 2631 . . . . . . . . 9 (𝑔 = 𝑒 → (𝑥 = (𝑔𝐶) ↔ 𝑥 = (𝑒𝐶)))
3635cbvrexv 3160 . . . . . . . 8 (∃𝑔𝐴 𝑥 = (𝑔𝐶) ↔ ∃𝑒𝐴 𝑥 = (𝑒𝐶))
37 sorpssi 6896 . . . . . . . . . . . . . . . 16 (( [] Or 𝐴 ∧ (𝑒𝐴𝑔𝐴)) → (𝑒𝑔𝑔𝑒))
38 ssdif 3723 . . . . . . . . . . . . . . . . 17 (𝑒𝑔 → (𝑒𝐶) ⊆ (𝑔𝐶))
39 ssdif 3723 . . . . . . . . . . . . . . . . 17 (𝑔𝑒 → (𝑔𝐶) ⊆ (𝑒𝐶))
4038, 39orim12i 538 . . . . . . . . . . . . . . . 16 ((𝑒𝑔𝑔𝑒) → ((𝑒𝐶) ⊆ (𝑔𝐶) ∨ (𝑔𝐶) ⊆ (𝑒𝐶)))
4137, 40syl 17 . . . . . . . . . . . . . . 15 (( [] Or 𝐴 ∧ (𝑒𝐴𝑔𝐴)) → ((𝑒𝐶) ⊆ (𝑔𝐶) ∨ (𝑔𝐶) ⊆ (𝑒𝐶)))
42 sseq2 3606 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑔𝐶) → ((𝑒𝐶) ⊆ 𝑓 ↔ (𝑒𝐶) ⊆ (𝑔𝐶)))
43 sseq1 3605 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑔𝐶) → (𝑓 ⊆ (𝑒𝐶) ↔ (𝑔𝐶) ⊆ (𝑒𝐶)))
4442, 43orbi12d 745 . . . . . . . . . . . . . . 15 (𝑓 = (𝑔𝐶) → (((𝑒𝐶) ⊆ 𝑓𝑓 ⊆ (𝑒𝐶)) ↔ ((𝑒𝐶) ⊆ (𝑔𝐶) ∨ (𝑔𝐶) ⊆ (𝑒𝐶))))
4541, 44syl5ibrcom 237 . . . . . . . . . . . . . 14 (( [] Or 𝐴 ∧ (𝑒𝐴𝑔𝐴)) → (𝑓 = (𝑔𝐶) → ((𝑒𝐶) ⊆ 𝑓𝑓 ⊆ (𝑒𝐶))))
4645expr 642 . . . . . . . . . . . . 13 (( [] Or 𝐴𝑒𝐴) → (𝑔𝐴 → (𝑓 = (𝑔𝐶) → ((𝑒𝐶) ⊆ 𝑓𝑓 ⊆ (𝑒𝐶)))))
4746rexlimdv 3023 . . . . . . . . . . . 12 (( [] Or 𝐴𝑒𝐴) → (∃𝑔𝐴 𝑓 = (𝑔𝐶) → ((𝑒𝐶) ⊆ 𝑓𝑓 ⊆ (𝑒𝐶))))
4812, 47syl5bi 232 . . . . . . . . . . 11 (( [] Or 𝐴𝑒𝐴) → (𝑓 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) → ((𝑒𝐶) ⊆ 𝑓𝑓 ⊆ (𝑒𝐶))))
4948ralrimiv 2959 . . . . . . . . . 10 (( [] Or 𝐴𝑒𝐴) → ∀𝑓 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶))((𝑒𝐶) ⊆ 𝑓𝑓 ⊆ (𝑒𝐶)))
50 sseq1 3605 . . . . . . . . . . . 12 (𝑥 = (𝑒𝐶) → (𝑥𝑓 ↔ (𝑒𝐶) ⊆ 𝑓))
51 sseq2 3606 . . . . . . . . . . . 12 (𝑥 = (𝑒𝐶) → (𝑓𝑥𝑓 ⊆ (𝑒𝐶)))
5250, 51orbi12d 745 . . . . . . . . . . 11 (𝑥 = (𝑒𝐶) → ((𝑥𝑓𝑓𝑥) ↔ ((𝑒𝐶) ⊆ 𝑓𝑓 ⊆ (𝑒𝐶))))
5352ralbidv 2980 . . . . . . . . . 10 (𝑥 = (𝑒𝐶) → (∀𝑓 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶))(𝑥𝑓𝑓𝑥) ↔ ∀𝑓 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶))((𝑒𝐶) ⊆ 𝑓𝑓 ⊆ (𝑒𝐶))))
5449, 53syl5ibrcom 237 . . . . . . . . 9 (( [] Or 𝐴𝑒𝐴) → (𝑥 = (𝑒𝐶) → ∀𝑓 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶))(𝑥𝑓𝑓𝑥)))
5554rexlimdva 3024 . . . . . . . 8 ( [] Or 𝐴 → (∃𝑒𝐴 𝑥 = (𝑒𝐶) → ∀𝑓 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶))(𝑥𝑓𝑓𝑥)))
5636, 55syl5bi 232 . . . . . . 7 ( [] Or 𝐴 → (∃𝑔𝐴 𝑥 = (𝑔𝐶) → ∀𝑓 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶))(𝑥𝑓𝑓𝑥)))
5733, 56syl5bi 232 . . . . . 6 ( [] Or 𝐴 → (𝑥 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) → ∀𝑓 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶))(𝑥𝑓𝑓𝑥)))
5857ralrimiv 2959 . . . . 5 ( [] Or 𝐴 → ∀𝑥 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶))∀𝑓 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶))(𝑥𝑓𝑓𝑥))
59 sorpss 6895 . . . . 5 ( [] Or ran (𝑔𝐴 ↦ (𝑔𝐶)) ↔ ∀𝑥 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶))∀𝑓 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶))(𝑥𝑓𝑓𝑥))
6058, 59sylibr 224 . . . 4 ( [] Or 𝐴 → [] Or ran (𝑔𝐴 ↦ (𝑔𝐶)))
6130, 60syl 17 . . 3 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) → [] Or ran (𝑔𝐴 ↦ (𝑔𝐶)))
62 fin2i 9061 . . 3 ((((𝐵𝐶) ∈ FinII ∧ ran (𝑔𝐴 ↦ (𝑔𝐶)) ⊆ 𝒫 (𝐵𝐶)) ∧ (ran (𝑔𝐴 ↦ (𝑔𝐶)) ≠ ∅ ∧ [] Or ran (𝑔𝐴 ↦ (𝑔𝐶)))) → ran (𝑔𝐴 ↦ (𝑔𝐶)) ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)))
631, 16, 29, 61, 62syl22anc 1324 . 2 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) → ran (𝑔𝐴 ↦ (𝑔𝐶)) ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)))
64 simpll3 1100 . . 3 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) → ¬ 𝐴𝐴)
65 difeq1 3699 . . . . . . 7 (𝑔 = 𝑓 → (𝑔𝐶) = (𝑓𝐶))
6665cbvmptv 4710 . . . . . 6 (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐴 ↦ (𝑓𝐶))
6766elrnmpt 5332 . . . . 5 ( ran (𝑔𝐴 ↦ (𝑔𝐶)) ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) → ( ran (𝑔𝐴 ↦ (𝑔𝐶)) ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) ↔ ∃𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)))
6867ibi 256 . . . 4 ( ran (𝑔𝐴 ↦ (𝑔𝐶)) ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) → ∃𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))
69 eqid 2621 . . . . . . . . . . . . . . . 16 (𝐶) = (𝐶)
70 difeq1 3699 . . . . . . . . . . . . . . . . . 18 (𝑔 = → (𝑔𝐶) = (𝐶))
7170eqeq2d 2631 . . . . . . . . . . . . . . . . 17 (𝑔 = → ((𝐶) = (𝑔𝐶) ↔ (𝐶) = (𝐶)))
7271rspcev 3295 . . . . . . . . . . . . . . . 16 ((𝐴 ∧ (𝐶) = (𝐶)) → ∃𝑔𝐴 (𝐶) = (𝑔𝐶))
7369, 72mpan2 706 . . . . . . . . . . . . . . 15 (𝐴 → ∃𝑔𝐴 (𝐶) = (𝑔𝐶))
7473adantl 482 . . . . . . . . . . . . . 14 (((𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝐴) → ∃𝑔𝐴 (𝐶) = (𝑔𝐶))
75 vex 3189 . . . . . . . . . . . . . . 15 ∈ V
76 difexg 4768 . . . . . . . . . . . . . . 15 ( ∈ V → (𝐶) ∈ V)
7710elrnmpt 5332 . . . . . . . . . . . . . . 15 ((𝐶) ∈ V → ((𝐶) ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) ↔ ∃𝑔𝐴 (𝐶) = (𝑔𝐶)))
7875, 76, 77mp2b 10 . . . . . . . . . . . . . 14 ((𝐶) ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) ↔ ∃𝑔𝐴 (𝐶) = (𝑔𝐶))
7974, 78sylibr 224 . . . . . . . . . . . . 13 (((𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝐴) → (𝐶) ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)))
80 elssuni 4433 . . . . . . . . . . . . 13 ((𝐶) ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) → (𝐶) ⊆ ran (𝑔𝐴 ↦ (𝑔𝐶)))
8179, 80syl 17 . . . . . . . . . . . 12 (((𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝐴) → (𝐶) ⊆ ran (𝑔𝐴 ↦ (𝑔𝐶)))
82 simplr 791 . . . . . . . . . . . 12 (((𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝐴) → ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))
8381, 82sseqtrd 3620 . . . . . . . . . . 11 (((𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝐴) → (𝐶) ⊆ (𝑓𝐶))
8483adantll 749 . . . . . . . . . 10 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → (𝐶) ⊆ (𝑓𝐶))
85 unss2 3762 . . . . . . . . . . 11 ((𝐶) ⊆ (𝑓𝐶) → (𝐶 ∪ (𝐶)) ⊆ (𝐶 ∪ (𝑓𝐶)))
86 uncom 3735 . . . . . . . . . . . . . . 15 (𝐶 ∪ (𝐶)) = ((𝐶) ∪ 𝐶)
87 undif1 4015 . . . . . . . . . . . . . . 15 ((𝐶) ∪ 𝐶) = (𝐶)
8886, 87eqtri 2643 . . . . . . . . . . . . . 14 (𝐶 ∪ (𝐶)) = (𝐶)
8988a1i 11 . . . . . . . . . . . . 13 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → (𝐶 ∪ (𝐶)) = (𝐶))
9064ad2antrr 761 . . . . . . . . . . . . . . . 16 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → ¬ 𝐴𝐴)
9117ad2antrr 761 . . . . . . . . . . . . . . . . 17 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → 𝐶𝐴)
92 simplrr 800 . . . . . . . . . . . . . . . . 17 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))
93 eqid 2621 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥𝐶) = (𝑥𝐶)
94 difeq1 3699 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑔 = 𝑥 → (𝑔𝐶) = (𝑥𝐶))
9594eqeq2d 2631 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑔 = 𝑥 → ((𝑥𝐶) = (𝑔𝐶) ↔ (𝑥𝐶) = (𝑥𝐶)))
9695rspcev 3295 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑥𝐴 ∧ (𝑥𝐶) = (𝑥𝐶)) → ∃𝑔𝐴 (𝑥𝐶) = (𝑔𝐶))
9793, 96mpan2 706 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥𝐴 → ∃𝑔𝐴 (𝑥𝐶) = (𝑔𝐶))
98 difexg 4768 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥 ∈ V → (𝑥𝐶) ∈ V)
9910elrnmpt 5332 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑥𝐶) ∈ V → ((𝑥𝐶) ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) ↔ ∃𝑔𝐴 (𝑥𝐶) = (𝑔𝐶)))
10031, 98, 99mp2b 10 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑥𝐶) ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) ↔ ∃𝑔𝐴 (𝑥𝐶) = (𝑔𝐶))
10197, 100sylibr 224 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥𝐴 → (𝑥𝐶) ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)))
102101adantl 482 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐶𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝑓𝐶) ∧ 𝑥𝐴) → (𝑥𝐶) ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)))
103 simpllr 798 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝐶𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝑓𝐶) ∧ 𝑥𝐴) → ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))
104 ssdif0 3916 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑓𝐶 ↔ (𝑓𝐶) = ∅)
105104biimpi 206 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑓𝐶 → (𝑓𝐶) = ∅)
106105ad2antlr 762 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝐶𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝑓𝐶) ∧ 𝑥𝐴) → (𝑓𝐶) = ∅)
107103, 106eqtrd 2655 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐶𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝑓𝐶) ∧ 𝑥𝐴) → ran (𝑔𝐴 ↦ (𝑔𝐶)) = ∅)
108 uni0c 4430 . . . . . . . . . . . . . . . . . . . . . . . . 25 ( ran (𝑔𝐴 ↦ (𝑔𝐶)) = ∅ ↔ ∀𝑒 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶))𝑒 = ∅)
109107, 108sylib 208 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐶𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝑓𝐶) ∧ 𝑥𝐴) → ∀𝑒 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶))𝑒 = ∅)
110 eqeq1 2625 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑒 = (𝑥𝐶) → (𝑒 = ∅ ↔ (𝑥𝐶) = ∅))
111110rspcva 3293 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑥𝐶) ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) ∧ ∀𝑒 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶))𝑒 = ∅) → (𝑥𝐶) = ∅)
112102, 109, 111syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐶𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝑓𝐶) ∧ 𝑥𝐴) → (𝑥𝐶) = ∅)
113 ssdif0 3916 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥𝐶 ↔ (𝑥𝐶) = ∅)
114112, 113sylibr 224 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐶𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝑓𝐶) ∧ 𝑥𝐴) → 𝑥𝐶)
115114ralrimiva 2960 . . . . . . . . . . . . . . . . . . . . 21 (((𝐶𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝑓𝐶) → ∀𝑥𝐴 𝑥𝐶)
116 unissb 4435 . . . . . . . . . . . . . . . . . . . . 21 ( 𝐴𝐶 ↔ ∀𝑥𝐴 𝑥𝐶)
117115, 116sylibr 224 . . . . . . . . . . . . . . . . . . . 20 (((𝐶𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝑓𝐶) → 𝐴𝐶)
118 elssuni 4433 . . . . . . . . . . . . . . . . . . . . 21 (𝐶𝐴𝐶 𝐴)
119118ad2antrr 761 . . . . . . . . . . . . . . . . . . . 20 (((𝐶𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝑓𝐶) → 𝐶 𝐴)
120117, 119eqssd 3600 . . . . . . . . . . . . . . . . . . 19 (((𝐶𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝑓𝐶) → 𝐴 = 𝐶)
121 simpll 789 . . . . . . . . . . . . . . . . . . 19 (((𝐶𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝑓𝐶) → 𝐶𝐴)
122120, 121eqeltrd 2698 . . . . . . . . . . . . . . . . . 18 (((𝐶𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝑓𝐶) → 𝐴𝐴)
123122ex 450 . . . . . . . . . . . . . . . . 17 ((𝐶𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) → (𝑓𝐶 𝐴𝐴))
12491, 92, 123syl2anc 692 . . . . . . . . . . . . . . . 16 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → (𝑓𝐶 𝐴𝐴))
12590, 124mtod 189 . . . . . . . . . . . . . . 15 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → ¬ 𝑓𝐶)
12630ad2antrr 761 . . . . . . . . . . . . . . . 16 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → [] Or 𝐴)
127 simplrl 799 . . . . . . . . . . . . . . . 16 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → 𝑓𝐴)
128 sorpssi 6896 . . . . . . . . . . . . . . . 16 (( [] Or 𝐴 ∧ (𝑓𝐴𝐶𝐴)) → (𝑓𝐶𝐶𝑓))
129126, 127, 91, 128syl12anc 1321 . . . . . . . . . . . . . . 15 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → (𝑓𝐶𝐶𝑓))
130 orel1 397 . . . . . . . . . . . . . . 15 𝑓𝐶 → ((𝑓𝐶𝐶𝑓) → 𝐶𝑓))
131125, 129, 130sylc 65 . . . . . . . . . . . . . 14 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → 𝐶𝑓)
132 undif 4021 . . . . . . . . . . . . . 14 (𝐶𝑓 ↔ (𝐶 ∪ (𝑓𝐶)) = 𝑓)
133131, 132sylib 208 . . . . . . . . . . . . 13 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → (𝐶 ∪ (𝑓𝐶)) = 𝑓)
13489, 133sseq12d 3613 . . . . . . . . . . . 12 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → ((𝐶 ∪ (𝐶)) ⊆ (𝐶 ∪ (𝑓𝐶)) ↔ (𝐶) ⊆ 𝑓))
135 ssun1 3754 . . . . . . . . . . . . 13 ⊆ (𝐶)
136 sstr 3591 . . . . . . . . . . . . 13 (( ⊆ (𝐶) ∧ (𝐶) ⊆ 𝑓) → 𝑓)
137135, 136mpan 705 . . . . . . . . . . . 12 ((𝐶) ⊆ 𝑓𝑓)
138134, 137syl6bi 243 . . . . . . . . . . 11 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → ((𝐶 ∪ (𝐶)) ⊆ (𝐶 ∪ (𝑓𝐶)) → 𝑓))
13985, 138syl5 34 . . . . . . . . . 10 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → ((𝐶) ⊆ (𝑓𝐶) → 𝑓))
14084, 139mpd 15 . . . . . . . . 9 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → 𝑓)
141140ralrimiva 2960 . . . . . . . 8 (((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) → ∀𝐴 𝑓)
142 unissb 4435 . . . . . . . 8 ( 𝐴𝑓 ↔ ∀𝐴 𝑓)
143141, 142sylibr 224 . . . . . . 7 (((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) → 𝐴𝑓)
144 elssuni 4433 . . . . . . . 8 (𝑓𝐴𝑓 𝐴)
145144ad2antrl 763 . . . . . . 7 (((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) → 𝑓 𝐴)
146143, 145eqssd 3600 . . . . . 6 (((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) → 𝐴 = 𝑓)
147 simprl 793 . . . . . 6 (((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) → 𝑓𝐴)
148146, 147eqeltrd 2698 . . . . 5 (((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) → 𝐴𝐴)
149148rexlimdvaa 3025 . . . 4 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) → (∃𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶) → 𝐴𝐴))
15068, 149syl5 34 . . 3 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) → ( ran (𝑔𝐴 ↦ (𝑔𝐶)) ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) → 𝐴𝐴))
15164, 150mtod 189 . 2 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) → ¬ ran (𝑔𝐴 ↦ (𝑔𝐶)) ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)))
15263, 151pm2.65da 599 1 (((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) → ¬ (𝐵𝐶) ∈ FinII)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  Vcvv 3186  cdif 3552  cun 3553  wss 3555  c0 3891  𝒫 cpw 4130   cuni 4402  cmpt 4673   Or wor 4994  ran crn 5075   [] crpss 6889  Fincfn 7899  FinIIcfin2 9045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-po 4995  df-so 4996  df-xp 5080  df-rel 5081  df-cnv 5082  df-dm 5084  df-rn 5085  df-rpss 6890  df-fin2 9052
This theorem is referenced by:  fin1a2s  9180
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