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Theorem fin1a2lem13 9487
Description: Lemma for fin1a2 9490. (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 1269 . . . 4 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) → 𝐴 ⊆ 𝒫 𝐵)
3 ssel2 3756 . . . . . . . . . 10 ((𝐴 ⊆ 𝒫 𝐵𝑔𝐴) → 𝑔 ∈ 𝒫 𝐵)
43elpwid 4327 . . . . . . . . 9 ((𝐴 ⊆ 𝒫 𝐵𝑔𝐴) → 𝑔𝐵)
54ssdifd 3908 . . . . . . . 8 ((𝐴 ⊆ 𝒫 𝐵𝑔𝐴) → (𝑔𝐶) ⊆ (𝐵𝐶))
6 sseq1 3786 . . . . . . . 8 (𝑓 = (𝑔𝐶) → (𝑓 ⊆ (𝐵𝐶) ↔ (𝑔𝐶) ⊆ (𝐵𝐶)))
75, 6syl5ibrcom 238 . . . . . . 7 ((𝐴 ⊆ 𝒫 𝐵𝑔𝐴) → (𝑓 = (𝑔𝐶) → 𝑓 ⊆ (𝐵𝐶)))
87rexlimdva 3178 . . . . . 6 (𝐴 ⊆ 𝒫 𝐵 → (∃𝑔𝐴 𝑓 = (𝑔𝐶) → 𝑓 ⊆ (𝐵𝐶)))
9 vex 3353 . . . . . . 7 𝑓 ∈ V
10 eqid 2765 . . . . . . . 8 (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑔𝐴 ↦ (𝑔𝐶))
1110elrnmpt 5541 . . . . . . 7 (𝑓 ∈ V → (𝑓 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) ↔ ∃𝑔𝐴 𝑓 = (𝑔𝐶)))
129, 11ax-mp 5 . . . . . 6 (𝑓 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) ↔ ∃𝑔𝐴 𝑓 = (𝑔𝐶))
13 selpw 4322 . . . . . 6 (𝑓 ∈ 𝒫 (𝐵𝐶) ↔ 𝑓 ⊆ (𝐵𝐶))
148, 12, 133imtr4g 287 . . . . 5 (𝐴 ⊆ 𝒫 𝐵 → (𝑓 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) → 𝑓 ∈ 𝒫 (𝐵𝐶)))
1514ssrdv 3767 . . . 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 796 . . . 4 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) → 𝐶𝐴)
18 difid 4113 . . . . . . 7 (𝐶𝐶) = ∅
1918eqcomi 2774 . . . . . 6 ∅ = (𝐶𝐶)
20 difeq1 3883 . . . . . . 7 (𝑔 = 𝐶 → (𝑔𝐶) = (𝐶𝐶))
2120rspceeqv 3479 . . . . . 6 ((𝐶𝐴 ∧ ∅ = (𝐶𝐶)) → ∃𝑔𝐴 ∅ = (𝑔𝐶))
2219, 21mpan2 682 . . . . 5 (𝐶𝐴 → ∃𝑔𝐴 ∅ = (𝑔𝐶))
23 0ex 4950 . . . . . 6 ∅ ∈ V
2410elrnmpt 5541 . . . . . 6 (∅ ∈ V → (∅ ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) ↔ ∃𝑔𝐴 ∅ = (𝑔𝐶)))
2523, 24ax-mp 5 . . . . 5 (∅ ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) ↔ ∃𝑔𝐴 ∅ = (𝑔𝐶))
2622, 25sylibr 225 . . . 4 (𝐶𝐴 → ∅ ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)))
27 ne0i 4085 . . . 4 (∅ ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) → ran (𝑔𝐴 ↦ (𝑔𝐶)) ≠ ∅)
2817, 26, 273syl 18 . . 3 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) → ran (𝑔𝐴 ↦ (𝑔𝐶)) ≠ ∅)
29 simpll2 1271 . . . 4 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) → [] Or 𝐴)
30 vex 3353 . . . . . . . 8 𝑥 ∈ V
3110elrnmpt 5541 . . . . . . . 8 (𝑥 ∈ V → (𝑥 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) ↔ ∃𝑔𝐴 𝑥 = (𝑔𝐶)))
3230, 31ax-mp 5 . . . . . . 7 (𝑥 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) ↔ ∃𝑔𝐴 𝑥 = (𝑔𝐶))
33 difeq1 3883 . . . . . . . . . 10 (𝑔 = 𝑒 → (𝑔𝐶) = (𝑒𝐶))
3433eqeq2d 2775 . . . . . . . . 9 (𝑔 = 𝑒 → (𝑥 = (𝑔𝐶) ↔ 𝑥 = (𝑒𝐶)))
3534cbvrexv 3320 . . . . . . . 8 (∃𝑔𝐴 𝑥 = (𝑔𝐶) ↔ ∃𝑒𝐴 𝑥 = (𝑒𝐶))
36 sorpssi 7141 . . . . . . . . . . . . . . . 16 (( [] Or 𝐴 ∧ (𝑒𝐴𝑔𝐴)) → (𝑒𝑔𝑔𝑒))
37 ssdif 3907 . . . . . . . . . . . . . . . . 17 (𝑒𝑔 → (𝑒𝐶) ⊆ (𝑔𝐶))
38 ssdif 3907 . . . . . . . . . . . . . . . . 17 (𝑔𝑒 → (𝑔𝐶) ⊆ (𝑒𝐶))
3937, 38orim12i 932 . . . . . . . . . . . . . . . 16 ((𝑒𝑔𝑔𝑒) → ((𝑒𝐶) ⊆ (𝑔𝐶) ∨ (𝑔𝐶) ⊆ (𝑒𝐶)))
4036, 39syl 17 . . . . . . . . . . . . . . 15 (( [] Or 𝐴 ∧ (𝑒𝐴𝑔𝐴)) → ((𝑒𝐶) ⊆ (𝑔𝐶) ∨ (𝑔𝐶) ⊆ (𝑒𝐶)))
41 sseq2 3787 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑔𝐶) → ((𝑒𝐶) ⊆ 𝑓 ↔ (𝑒𝐶) ⊆ (𝑔𝐶)))
42 sseq1 3786 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑔𝐶) → (𝑓 ⊆ (𝑒𝐶) ↔ (𝑔𝐶) ⊆ (𝑒𝐶)))
4341, 42orbi12d 942 . . . . . . . . . . . . . . 15 (𝑓 = (𝑔𝐶) → (((𝑒𝐶) ⊆ 𝑓𝑓 ⊆ (𝑒𝐶)) ↔ ((𝑒𝐶) ⊆ (𝑔𝐶) ∨ (𝑔𝐶) ⊆ (𝑒𝐶))))
4440, 43syl5ibrcom 238 . . . . . . . . . . . . . 14 (( [] Or 𝐴 ∧ (𝑒𝐴𝑔𝐴)) → (𝑓 = (𝑔𝐶) → ((𝑒𝐶) ⊆ 𝑓𝑓 ⊆ (𝑒𝐶))))
4544expr 448 . . . . . . . . . . . . 13 (( [] Or 𝐴𝑒𝐴) → (𝑔𝐴 → (𝑓 = (𝑔𝐶) → ((𝑒𝐶) ⊆ 𝑓𝑓 ⊆ (𝑒𝐶)))))
4645rexlimdv 3177 . . . . . . . . . . . 12 (( [] Or 𝐴𝑒𝐴) → (∃𝑔𝐴 𝑓 = (𝑔𝐶) → ((𝑒𝐶) ⊆ 𝑓𝑓 ⊆ (𝑒𝐶))))
4712, 46syl5bi 233 . . . . . . . . . . 11 (( [] Or 𝐴𝑒𝐴) → (𝑓 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) → ((𝑒𝐶) ⊆ 𝑓𝑓 ⊆ (𝑒𝐶))))
4847ralrimiv 3112 . . . . . . . . . 10 (( [] Or 𝐴𝑒𝐴) → ∀𝑓 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶))((𝑒𝐶) ⊆ 𝑓𝑓 ⊆ (𝑒𝐶)))
49 sseq1 3786 . . . . . . . . . . . 12 (𝑥 = (𝑒𝐶) → (𝑥𝑓 ↔ (𝑒𝐶) ⊆ 𝑓))
50 sseq2 3787 . . . . . . . . . . . 12 (𝑥 = (𝑒𝐶) → (𝑓𝑥𝑓 ⊆ (𝑒𝐶)))
5149, 50orbi12d 942 . . . . . . . . . . 11 (𝑥 = (𝑒𝐶) → ((𝑥𝑓𝑓𝑥) ↔ ((𝑒𝐶) ⊆ 𝑓𝑓 ⊆ (𝑒𝐶))))
5251ralbidv 3133 . . . . . . . . . 10 (𝑥 = (𝑒𝐶) → (∀𝑓 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶))(𝑥𝑓𝑓𝑥) ↔ ∀𝑓 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶))((𝑒𝐶) ⊆ 𝑓𝑓 ⊆ (𝑒𝐶))))
5348, 52syl5ibrcom 238 . . . . . . . . 9 (( [] Or 𝐴𝑒𝐴) → (𝑥 = (𝑒𝐶) → ∀𝑓 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶))(𝑥𝑓𝑓𝑥)))
5453rexlimdva 3178 . . . . . . . 8 ( [] Or 𝐴 → (∃𝑒𝐴 𝑥 = (𝑒𝐶) → ∀𝑓 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶))(𝑥𝑓𝑓𝑥)))
5535, 54syl5bi 233 . . . . . . 7 ( [] Or 𝐴 → (∃𝑔𝐴 𝑥 = (𝑔𝐶) → ∀𝑓 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶))(𝑥𝑓𝑓𝑥)))
5632, 55syl5bi 233 . . . . . 6 ( [] Or 𝐴 → (𝑥 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) → ∀𝑓 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶))(𝑥𝑓𝑓𝑥)))
5756ralrimiv 3112 . . . . 5 ( [] Or 𝐴 → ∀𝑥 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶))∀𝑓 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶))(𝑥𝑓𝑓𝑥))
58 sorpss 7140 . . . . 5 ( [] Or ran (𝑔𝐴 ↦ (𝑔𝐶)) ↔ ∀𝑥 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶))∀𝑓 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶))(𝑥𝑓𝑓𝑥))
5957, 58sylibr 225 . . . 4 ( [] Or 𝐴 → [] Or ran (𝑔𝐴 ↦ (𝑔𝐶)))
6029, 59syl 17 . . 3 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) → [] Or ran (𝑔𝐴 ↦ (𝑔𝐶)))
61 fin2i 9370 . . 3 ((((𝐵𝐶) ∈ FinII ∧ ran (𝑔𝐴 ↦ (𝑔𝐶)) ⊆ 𝒫 (𝐵𝐶)) ∧ (ran (𝑔𝐴 ↦ (𝑔𝐶)) ≠ ∅ ∧ [] Or ran (𝑔𝐴 ↦ (𝑔𝐶)))) → ran (𝑔𝐴 ↦ (𝑔𝐶)) ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)))
621, 16, 28, 60, 61syl22anc 867 . 2 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) → ran (𝑔𝐴 ↦ (𝑔𝐶)) ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)))
63 simpll3 1273 . . 3 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) → ¬ 𝐴𝐴)
64 difeq1 3883 . . . . . . 7 (𝑔 = 𝑓 → (𝑔𝐶) = (𝑓𝐶))
6564cbvmptv 4909 . . . . . 6 (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐴 ↦ (𝑓𝐶))
6665elrnmpt 5541 . . . . 5 ( ran (𝑔𝐴 ↦ (𝑔𝐶)) ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) → ( ran (𝑔𝐴 ↦ (𝑔𝐶)) ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) ↔ ∃𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)))
6766ibi 258 . . . 4 ( ran (𝑔𝐴 ↦ (𝑔𝐶)) ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) → ∃𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))
68 eqid 2765 . . . . . . . . . . . . . . . 16 (𝐶) = (𝐶)
69 difeq1 3883 . . . . . . . . . . . . . . . . 17 (𝑔 = → (𝑔𝐶) = (𝐶))
7069rspceeqv 3479 . . . . . . . . . . . . . . . 16 ((𝐴 ∧ (𝐶) = (𝐶)) → ∃𝑔𝐴 (𝐶) = (𝑔𝐶))
7168, 70mpan2 682 . . . . . . . . . . . . . . 15 (𝐴 → ∃𝑔𝐴 (𝐶) = (𝑔𝐶))
7271adantl 473 . . . . . . . . . . . . . 14 (((𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝐴) → ∃𝑔𝐴 (𝐶) = (𝑔𝐶))
73 vex 3353 . . . . . . . . . . . . . . 15 ∈ V
74 difexg 4969 . . . . . . . . . . . . . . 15 ( ∈ V → (𝐶) ∈ V)
7510elrnmpt 5541 . . . . . . . . . . . . . . 15 ((𝐶) ∈ V → ((𝐶) ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) ↔ ∃𝑔𝐴 (𝐶) = (𝑔𝐶)))
7673, 74, 75mp2b 10 . . . . . . . . . . . . . 14 ((𝐶) ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) ↔ ∃𝑔𝐴 (𝐶) = (𝑔𝐶))
7772, 76sylibr 225 . . . . . . . . . . . . 13 (((𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝐴) → (𝐶) ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)))
78 elssuni 4625 . . . . . . . . . . . . 13 ((𝐶) ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) → (𝐶) ⊆ ran (𝑔𝐴 ↦ (𝑔𝐶)))
7977, 78syl 17 . . . . . . . . . . . 12 (((𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝐴) → (𝐶) ⊆ ran (𝑔𝐴 ↦ (𝑔𝐶)))
80 simplr 785 . . . . . . . . . . . 12 (((𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝐴) → ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))
8179, 80sseqtrd 3801 . . . . . . . . . . 11 (((𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝐴) → (𝐶) ⊆ (𝑓𝐶))
8281adantll 705 . . . . . . . . . 10 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → (𝐶) ⊆ (𝑓𝐶))
83 unss2 3946 . . . . . . . . . . 11 ((𝐶) ⊆ (𝑓𝐶) → (𝐶 ∪ (𝐶)) ⊆ (𝐶 ∪ (𝑓𝐶)))
84 uncom 3919 . . . . . . . . . . . . . . 15 (𝐶 ∪ (𝐶)) = ((𝐶) ∪ 𝐶)
85 undif1 4203 . . . . . . . . . . . . . . 15 ((𝐶) ∪ 𝐶) = (𝐶)
8684, 85eqtri 2787 . . . . . . . . . . . . . 14 (𝐶 ∪ (𝐶)) = (𝐶)
8786a1i 11 . . . . . . . . . . . . 13 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → (𝐶 ∪ (𝐶)) = (𝐶))
8863ad2antrr 717 . . . . . . . . . . . . . . . 16 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → ¬ 𝐴𝐴)
8917ad2antrr 717 . . . . . . . . . . . . . . . . 17 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → 𝐶𝐴)
90 simplrr 796 . . . . . . . . . . . . . . . . 17 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))
91 eqeq1 2769 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑒 = (𝑥𝐶) → (𝑒 = ∅ ↔ (𝑥𝐶) = ∅))
92 simpllr 793 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝐶𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝑓𝐶) ∧ 𝑥𝐴) → ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))
93 ssdif0 4106 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑓𝐶 ↔ (𝑓𝐶) = ∅)
9493biimpi 207 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑓𝐶 → (𝑓𝐶) = ∅)
9594ad2antlr 718 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝐶𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝑓𝐶) ∧ 𝑥𝐴) → (𝑓𝐶) = ∅)
9692, 95eqtrd 2799 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐶𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝑓𝐶) ∧ 𝑥𝐴) → ran (𝑔𝐴 ↦ (𝑔𝐶)) = ∅)
97 uni0c 4622 . . . . . . . . . . . . . . . . . . . . . . . . 25 ( ran (𝑔𝐴 ↦ (𝑔𝐶)) = ∅ ↔ ∀𝑒 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶))𝑒 = ∅)
9896, 97sylib 209 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐶𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝑓𝐶) ∧ 𝑥𝐴) → ∀𝑒 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶))𝑒 = ∅)
99 eqid 2765 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥𝐶) = (𝑥𝐶)
100 difeq1 3883 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑔 = 𝑥 → (𝑔𝐶) = (𝑥𝐶))
101100rspceeqv 3479 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑥𝐴 ∧ (𝑥𝐶) = (𝑥𝐶)) → ∃𝑔𝐴 (𝑥𝐶) = (𝑔𝐶))
10299, 101mpan2 682 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥𝐴 → ∃𝑔𝐴 (𝑥𝐶) = (𝑔𝐶))
103 difexg 4969 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥 ∈ V → (𝑥𝐶) ∈ V)
10410elrnmpt 5541 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑥𝐶) ∈ V → ((𝑥𝐶) ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) ↔ ∃𝑔𝐴 (𝑥𝐶) = (𝑔𝐶)))
10530, 103, 104mp2b 10 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑥𝐶) ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) ↔ ∃𝑔𝐴 (𝑥𝐶) = (𝑔𝐶))
106102, 105sylibr 225 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥𝐴 → (𝑥𝐶) ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)))
107106adantl 473 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐶𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝑓𝐶) ∧ 𝑥𝐴) → (𝑥𝐶) ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)))
10891, 98, 107rspcdva 3467 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐶𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝑓𝐶) ∧ 𝑥𝐴) → (𝑥𝐶) = ∅)
109 ssdif0 4106 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥𝐶 ↔ (𝑥𝐶) = ∅)
110108, 109sylibr 225 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐶𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝑓𝐶) ∧ 𝑥𝐴) → 𝑥𝐶)
111110ralrimiva 3113 . . . . . . . . . . . . . . . . . . . . 21 (((𝐶𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝑓𝐶) → ∀𝑥𝐴 𝑥𝐶)
112 unissb 4627 . . . . . . . . . . . . . . . . . . . . 21 ( 𝐴𝐶 ↔ ∀𝑥𝐴 𝑥𝐶)
113111, 112sylibr 225 . . . . . . . . . . . . . . . . . . . 20 (((𝐶𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝑓𝐶) → 𝐴𝐶)
114 elssuni 4625 . . . . . . . . . . . . . . . . . . . . 21 (𝐶𝐴𝐶 𝐴)
115114ad2antrr 717 . . . . . . . . . . . . . . . . . . . 20 (((𝐶𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝑓𝐶) → 𝐶 𝐴)
116113, 115eqssd 3778 . . . . . . . . . . . . . . . . . . 19 (((𝐶𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝑓𝐶) → 𝐴 = 𝐶)
117 simpll 783 . . . . . . . . . . . . . . . . . . 19 (((𝐶𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝑓𝐶) → 𝐶𝐴)
118116, 117eqeltrd 2844 . . . . . . . . . . . . . . . . . 18 (((𝐶𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝑓𝐶) → 𝐴𝐴)
119118ex 401 . . . . . . . . . . . . . . . . 17 ((𝐶𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) → (𝑓𝐶 𝐴𝐴))
12089, 90, 119syl2anc 579 . . . . . . . . . . . . . . . 16 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → (𝑓𝐶 𝐴𝐴))
12188, 120mtod 189 . . . . . . . . . . . . . . 15 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → ¬ 𝑓𝐶)
12229ad2antrr 717 . . . . . . . . . . . . . . . 16 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → [] Or 𝐴)
123 simplrl 795 . . . . . . . . . . . . . . . 16 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → 𝑓𝐴)
124 sorpssi 7141 . . . . . . . . . . . . . . . 16 (( [] Or 𝐴 ∧ (𝑓𝐴𝐶𝐴)) → (𝑓𝐶𝐶𝑓))
125122, 123, 89, 124syl12anc 865 . . . . . . . . . . . . . . 15 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → (𝑓𝐶𝐶𝑓))
126 orel1 912 . . . . . . . . . . . . . . 15 𝑓𝐶 → ((𝑓𝐶𝐶𝑓) → 𝐶𝑓))
127121, 125, 126sylc 65 . . . . . . . . . . . . . 14 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → 𝐶𝑓)
128 undif 4209 . . . . . . . . . . . . . 14 (𝐶𝑓 ↔ (𝐶 ∪ (𝑓𝐶)) = 𝑓)
129127, 128sylib 209 . . . . . . . . . . . . 13 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → (𝐶 ∪ (𝑓𝐶)) = 𝑓)
13087, 129sseq12d 3794 . . . . . . . . . . . 12 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → ((𝐶 ∪ (𝐶)) ⊆ (𝐶 ∪ (𝑓𝐶)) ↔ (𝐶) ⊆ 𝑓))
131 ssun1 3938 . . . . . . . . . . . . 13 ⊆ (𝐶)
132 sstr 3769 . . . . . . . . . . . . 13 (( ⊆ (𝐶) ∧ (𝐶) ⊆ 𝑓) → 𝑓)
133131, 132mpan 681 . . . . . . . . . . . 12 ((𝐶) ⊆ 𝑓𝑓)
134130, 133syl6bi 244 . . . . . . . . . . 11 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → ((𝐶 ∪ (𝐶)) ⊆ (𝐶 ∪ (𝑓𝐶)) → 𝑓))
13583, 134syl5 34 . . . . . . . . . 10 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → ((𝐶) ⊆ (𝑓𝐶) → 𝑓))
13682, 135mpd 15 . . . . . . . . 9 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → 𝑓)
137136ralrimiva 3113 . . . . . . . 8 (((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) → ∀𝐴 𝑓)
138 unissb 4627 . . . . . . . 8 ( 𝐴𝑓 ↔ ∀𝐴 𝑓)
139137, 138sylibr 225 . . . . . . 7 (((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) → 𝐴𝑓)
140 elssuni 4625 . . . . . . . 8 (𝑓𝐴𝑓 𝐴)
141140ad2antrl 719 . . . . . . 7 (((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) → 𝑓 𝐴)
142139, 141eqssd 3778 . . . . . 6 (((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) → 𝐴 = 𝑓)
143 simprl 787 . . . . . 6 (((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) → 𝑓𝐴)
144142, 143eqeltrd 2844 . . . . 5 (((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) → 𝐴𝐴)
145144rexlimdvaa 3179 . . . 4 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) → (∃𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶) → 𝐴𝐴))
14667, 145syl5 34 . . 3 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) → ( ran (𝑔𝐴 ↦ (𝑔𝐶)) ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) → 𝐴𝐴))
14763, 146mtod 189 . 2 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) → ¬ ran (𝑔𝐴 ↦ (𝑔𝐶)) ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)))
14862, 147pm2.65da 851 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 197  wa 384  wo 873  w3a 1107   = wceq 1652  wcel 2155  wne 2937  wral 3055  wrex 3056  Vcvv 3350  cdif 3729  cun 3730  wss 3732  c0 4079  𝒫 cpw 4315   cuni 4594  cmpt 4888   Or wor 5197  ran crn 5278   [] crpss 7134  Fincfn 8160  FinIIcfin2 9354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-po 5198  df-so 5199  df-xp 5283  df-rel 5284  df-cnv 5285  df-dm 5287  df-rn 5288  df-rpss 7135  df-fin2 9361
This theorem is referenced by:  fin1a2s  9489
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