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Theorem hashf1lem1OLD 14354
Description: Obsolete version of hashf1lem1 14353 as of 7-Aug-2024. (Contributed by Mario Carneiro, 17-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
hashf1lem2.1 (𝜑𝐴 ∈ Fin)
hashf1lem2.2 (𝜑𝐵 ∈ Fin)
hashf1lem2.3 (𝜑 → ¬ 𝑧𝐴)
hashf1lem2.4 (𝜑 → ((♯‘𝐴) + 1) ≤ (♯‘𝐵))
hashf1lem1.5 (𝜑𝐹:𝐴1-1𝐵)
Assertion
Ref Expression
hashf1lem1OLD (𝜑 → {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ≈ (𝐵 ∖ ran 𝐹))
Distinct variable groups:   𝑧,𝑓   𝐴,𝑓   𝐵,𝑓   𝜑,𝑓   𝑓,𝐹
Allowed substitution hints:   𝜑(𝑧)   𝐴(𝑧)   𝐵(𝑧)   𝐹(𝑧)

Proof of Theorem hashf1lem1OLD
Dummy variables 𝑔 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1f 6738 . . . . . 6 (𝑓:(𝐴 ∪ {𝑧})–1-1𝐵𝑓:(𝐴 ∪ {𝑧})⟶𝐵)
21adantl 482 . . . . 5 (((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵) → 𝑓:(𝐴 ∪ {𝑧})⟶𝐵)
3 hashf1lem2.2 . . . . . 6 (𝜑𝐵 ∈ Fin)
4 hashf1lem2.1 . . . . . . 7 (𝜑𝐴 ∈ Fin)
5 snfi 8988 . . . . . . 7 {𝑧} ∈ Fin
6 unfi 9116 . . . . . . 7 ((𝐴 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝐴 ∪ {𝑧}) ∈ Fin)
74, 5, 6sylancl 586 . . . . . 6 (𝜑 → (𝐴 ∪ {𝑧}) ∈ Fin)
83, 7elmapd 8779 . . . . 5 (𝜑 → (𝑓 ∈ (𝐵m (𝐴 ∪ {𝑧})) ↔ 𝑓:(𝐴 ∪ {𝑧})⟶𝐵))
92, 8syl5ibr 245 . . . 4 (𝜑 → (((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵) → 𝑓 ∈ (𝐵m (𝐴 ∪ {𝑧}))))
109abssdv 4025 . . 3 (𝜑 → {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ⊆ (𝐵m (𝐴 ∪ {𝑧})))
11 ovex 7390 . . 3 (𝐵m (𝐴 ∪ {𝑧})) ∈ V
12 ssexg 5280 . . 3 (({𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ⊆ (𝐵m (𝐴 ∪ {𝑧})) ∧ (𝐵m (𝐴 ∪ {𝑧})) ∈ V) → {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ∈ V)
1310, 11, 12sylancl 586 . 2 (𝜑 → {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ∈ V)
14 difexg 5284 . . 3 (𝐵 ∈ Fin → (𝐵 ∖ ran 𝐹) ∈ V)
153, 14syl 17 . 2 (𝜑 → (𝐵 ∖ ran 𝐹) ∈ V)
16 vex 3449 . . . 4 𝑔 ∈ V
17 reseq1 5931 . . . . . 6 (𝑓 = 𝑔 → (𝑓𝐴) = (𝑔𝐴))
1817eqeq1d 2738 . . . . 5 (𝑓 = 𝑔 → ((𝑓𝐴) = 𝐹 ↔ (𝑔𝐴) = 𝐹))
19 f1eq1 6733 . . . . 5 (𝑓 = 𝑔 → (𝑓:(𝐴 ∪ {𝑧})–1-1𝐵𝑔:(𝐴 ∪ {𝑧})–1-1𝐵))
2018, 19anbi12d 631 . . . 4 (𝑓 = 𝑔 → (((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵) ↔ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)))
2116, 20elab 3630 . . 3 (𝑔 ∈ {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ↔ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵))
22 f1f 6738 . . . . . . 7 (𝑔:(𝐴 ∪ {𝑧})–1-1𝐵𝑔:(𝐴 ∪ {𝑧})⟶𝐵)
2322ad2antll 727 . . . . . 6 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → 𝑔:(𝐴 ∪ {𝑧})⟶𝐵)
24 ssun2 4133 . . . . . . 7 {𝑧} ⊆ (𝐴 ∪ {𝑧})
25 vex 3449 . . . . . . . 8 𝑧 ∈ V
2625snss 4746 . . . . . . 7 (𝑧 ∈ (𝐴 ∪ {𝑧}) ↔ {𝑧} ⊆ (𝐴 ∪ {𝑧}))
2724, 26mpbir 230 . . . . . 6 𝑧 ∈ (𝐴 ∪ {𝑧})
28 ffvelcdm 7032 . . . . . 6 ((𝑔:(𝐴 ∪ {𝑧})⟶𝐵𝑧 ∈ (𝐴 ∪ {𝑧})) → (𝑔𝑧) ∈ 𝐵)
2923, 27, 28sylancl 586 . . . . 5 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → (𝑔𝑧) ∈ 𝐵)
30 hashf1lem2.3 . . . . . . 7 (𝜑 → ¬ 𝑧𝐴)
3130adantr 481 . . . . . 6 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → ¬ 𝑧𝐴)
32 df-ima 5646 . . . . . . . . 9 (𝑔𝐴) = ran (𝑔𝐴)
33 simprl 769 . . . . . . . . . 10 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → (𝑔𝐴) = 𝐹)
3433rneqd 5893 . . . . . . . . 9 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → ran (𝑔𝐴) = ran 𝐹)
3532, 34eqtrid 2788 . . . . . . . 8 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → (𝑔𝐴) = ran 𝐹)
3635eleq2d 2823 . . . . . . 7 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → ((𝑔𝑧) ∈ (𝑔𝐴) ↔ (𝑔𝑧) ∈ ran 𝐹))
37 simprr 771 . . . . . . . 8 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → 𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)
3827a1i 11 . . . . . . . 8 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → 𝑧 ∈ (𝐴 ∪ {𝑧}))
39 ssun1 4132 . . . . . . . . 9 𝐴 ⊆ (𝐴 ∪ {𝑧})
4039a1i 11 . . . . . . . 8 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → 𝐴 ⊆ (𝐴 ∪ {𝑧}))
41 f1elima 7210 . . . . . . . 8 ((𝑔:(𝐴 ∪ {𝑧})–1-1𝐵𝑧 ∈ (𝐴 ∪ {𝑧}) ∧ 𝐴 ⊆ (𝐴 ∪ {𝑧})) → ((𝑔𝑧) ∈ (𝑔𝐴) ↔ 𝑧𝐴))
4237, 38, 40, 41syl3anc 1371 . . . . . . 7 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → ((𝑔𝑧) ∈ (𝑔𝐴) ↔ 𝑧𝐴))
4336, 42bitr3d 280 . . . . . 6 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → ((𝑔𝑧) ∈ ran 𝐹𝑧𝐴))
4431, 43mtbird 324 . . . . 5 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → ¬ (𝑔𝑧) ∈ ran 𝐹)
4529, 44eldifd 3921 . . . 4 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → (𝑔𝑧) ∈ (𝐵 ∖ ran 𝐹))
4645ex 413 . . 3 (𝜑 → (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) → (𝑔𝑧) ∈ (𝐵 ∖ ran 𝐹)))
4721, 46biimtrid 241 . 2 (𝜑 → (𝑔 ∈ {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} → (𝑔𝑧) ∈ (𝐵 ∖ ran 𝐹)))
48 hashf1lem1.5 . . . . . . 7 (𝜑𝐹:𝐴1-1𝐵)
49 f1f 6738 . . . . . . 7 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
5048, 49syl 17 . . . . . 6 (𝜑𝐹:𝐴𝐵)
5150adantr 481 . . . . 5 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → 𝐹:𝐴𝐵)
52 vex 3449 . . . . . . . 8 𝑥 ∈ V
5325, 52f1osn 6824 . . . . . . 7 {⟨𝑧, 𝑥⟩}:{𝑧}–1-1-onto→{𝑥}
54 f1of 6784 . . . . . . 7 ({⟨𝑧, 𝑥⟩}:{𝑧}–1-1-onto→{𝑥} → {⟨𝑧, 𝑥⟩}:{𝑧}⟶{𝑥})
5553, 54ax-mp 5 . . . . . 6 {⟨𝑧, 𝑥⟩}:{𝑧}⟶{𝑥}
56 eldifi 4086 . . . . . . . 8 (𝑥 ∈ (𝐵 ∖ ran 𝐹) → 𝑥𝐵)
5756adantl 482 . . . . . . 7 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → 𝑥𝐵)
5857snssd 4769 . . . . . 6 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → {𝑥} ⊆ 𝐵)
59 fss 6685 . . . . . 6 (({⟨𝑧, 𝑥⟩}:{𝑧}⟶{𝑥} ∧ {𝑥} ⊆ 𝐵) → {⟨𝑧, 𝑥⟩}:{𝑧}⟶𝐵)
6055, 58, 59sylancr 587 . . . . 5 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → {⟨𝑧, 𝑥⟩}:{𝑧}⟶𝐵)
61 res0 5941 . . . . . . 7 (𝐹 ↾ ∅) = ∅
62 res0 5941 . . . . . . 7 ({⟨𝑧, 𝑥⟩} ↾ ∅) = ∅
6361, 62eqtr4i 2767 . . . . . 6 (𝐹 ↾ ∅) = ({⟨𝑧, 𝑥⟩} ↾ ∅)
64 disjsn 4672 . . . . . . . . 9 ((𝐴 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝐴)
6530, 64sylibr 233 . . . . . . . 8 (𝜑 → (𝐴 ∩ {𝑧}) = ∅)
6665adantr 481 . . . . . . 7 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐴 ∩ {𝑧}) = ∅)
6766reseq2d 5937 . . . . . 6 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ↾ (𝐴 ∩ {𝑧})) = (𝐹 ↾ ∅))
6866reseq2d 5937 . . . . . 6 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → ({⟨𝑧, 𝑥⟩} ↾ (𝐴 ∩ {𝑧})) = ({⟨𝑧, 𝑥⟩} ↾ ∅))
6963, 67, 683eqtr4a 2802 . . . . 5 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ↾ (𝐴 ∩ {𝑧})) = ({⟨𝑧, 𝑥⟩} ↾ (𝐴 ∩ {𝑧})))
70 fresaunres1 6715 . . . . 5 ((𝐹:𝐴𝐵 ∧ {⟨𝑧, 𝑥⟩}:{𝑧}⟶𝐵 ∧ (𝐹 ↾ (𝐴 ∩ {𝑧})) = ({⟨𝑧, 𝑥⟩} ↾ (𝐴 ∩ {𝑧}))) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴) = 𝐹)
7151, 60, 69, 70syl3anc 1371 . . . 4 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴) = 𝐹)
72 f1f1orn 6795 . . . . . . . . 9 (𝐹:𝐴1-1𝐵𝐹:𝐴1-1-onto→ran 𝐹)
7348, 72syl 17 . . . . . . . 8 (𝜑𝐹:𝐴1-1-onto→ran 𝐹)
7473adantr 481 . . . . . . 7 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → 𝐹:𝐴1-1-onto→ran 𝐹)
7553a1i 11 . . . . . . 7 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → {⟨𝑧, 𝑥⟩}:{𝑧}–1-1-onto→{𝑥})
76 eldifn 4087 . . . . . . . . 9 (𝑥 ∈ (𝐵 ∖ ran 𝐹) → ¬ 𝑥 ∈ ran 𝐹)
7776adantl 482 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → ¬ 𝑥 ∈ ran 𝐹)
78 disjsn 4672 . . . . . . . 8 ((ran 𝐹 ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ ran 𝐹)
7977, 78sylibr 233 . . . . . . 7 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (ran 𝐹 ∩ {𝑥}) = ∅)
80 f1oun 6803 . . . . . . 7 (((𝐹:𝐴1-1-onto→ran 𝐹 ∧ {⟨𝑧, 𝑥⟩}:{𝑧}–1-1-onto→{𝑥}) ∧ ((𝐴 ∩ {𝑧}) = ∅ ∧ (ran 𝐹 ∩ {𝑥}) = ∅)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1-onto→(ran 𝐹 ∪ {𝑥}))
8174, 75, 66, 79, 80syl22anc 837 . . . . . 6 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1-onto→(ran 𝐹 ∪ {𝑥}))
82 f1of1 6783 . . . . . 6 ((𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1-onto→(ran 𝐹 ∪ {𝑥}) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1→(ran 𝐹 ∪ {𝑥}))
8381, 82syl 17 . . . . 5 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1→(ran 𝐹 ∪ {𝑥}))
8451frnd 6676 . . . . . 6 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → ran 𝐹𝐵)
8584, 58unssd 4146 . . . . 5 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (ran 𝐹 ∪ {𝑥}) ⊆ 𝐵)
86 f1ss 6744 . . . . 5 (((𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1→(ran 𝐹 ∪ {𝑥}) ∧ (ran 𝐹 ∪ {𝑥}) ⊆ 𝐵) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1𝐵)
8783, 85, 86syl2anc 584 . . . 4 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1𝐵)
8850, 4fexd 7177 . . . . . . 7 (𝜑𝐹 ∈ V)
8988adantr 481 . . . . . 6 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → 𝐹 ∈ V)
90 snex 5388 . . . . . 6 {⟨𝑧, 𝑥⟩} ∈ V
91 unexg 7683 . . . . . 6 ((𝐹 ∈ V ∧ {⟨𝑧, 𝑥⟩} ∈ V) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ V)
9289, 90, 91sylancl 586 . . . . 5 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ V)
93 reseq1 5931 . . . . . . . 8 (𝑓 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) → (𝑓𝐴) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴))
9493eqeq1d 2738 . . . . . . 7 (𝑓 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) → ((𝑓𝐴) = 𝐹 ↔ ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴) = 𝐹))
95 f1eq1 6733 . . . . . . 7 (𝑓 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) → (𝑓:(𝐴 ∪ {𝑧})–1-1𝐵 ↔ (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1𝐵))
9694, 95anbi12d 631 . . . . . 6 (𝑓 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) → (((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵) ↔ (((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴) = 𝐹 ∧ (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1𝐵)))
9796elabg 3628 . . . . 5 ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ V → ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ↔ (((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴) = 𝐹 ∧ (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1𝐵)))
9892, 97syl 17 . . . 4 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ↔ (((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴) = 𝐹 ∧ (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1𝐵)))
9971, 87, 98mpbir2and 711 . . 3 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)})
10099ex 413 . 2 (𝜑 → (𝑥 ∈ (𝐵 ∖ ran 𝐹) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)}))
10121anbi1i 624 . . 3 ((𝑔 ∈ {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) ↔ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)))
102 simprlr 778 . . . . . . 7 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → 𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)
103 f1fn 6739 . . . . . . 7 (𝑔:(𝐴 ∪ {𝑧})–1-1𝐵𝑔 Fn (𝐴 ∪ {𝑧}))
104102, 103syl 17 . . . . . 6 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → 𝑔 Fn (𝐴 ∪ {𝑧}))
10581adantrl 714 . . . . . . 7 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1-onto→(ran 𝐹 ∪ {𝑥}))
106 f1ofn 6785 . . . . . . 7 ((𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1-onto→(ran 𝐹 ∪ {𝑥}) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) Fn (𝐴 ∪ {𝑧}))
107105, 106syl 17 . . . . . 6 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) Fn (𝐴 ∪ {𝑧}))
108 eqfnfv 6982 . . . . . 6 ((𝑔 Fn (𝐴 ∪ {𝑧}) ∧ (𝐹 ∪ {⟨𝑧, 𝑥⟩}) Fn (𝐴 ∪ {𝑧})) → (𝑔 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↔ ∀𝑦 ∈ (𝐴 ∪ {𝑧})(𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦)))
109104, 107, 108syl2anc 584 . . . . 5 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (𝑔 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↔ ∀𝑦 ∈ (𝐴 ∪ {𝑧})(𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦)))
110 fvres 6861 . . . . . . . . . . 11 (𝑦𝐴 → ((𝑔𝐴)‘𝑦) = (𝑔𝑦))
111110eqcomd 2742 . . . . . . . . . 10 (𝑦𝐴 → (𝑔𝑦) = ((𝑔𝐴)‘𝑦))
112 simprll 777 . . . . . . . . . . 11 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (𝑔𝐴) = 𝐹)
113112fveq1d 6844 . . . . . . . . . 10 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → ((𝑔𝐴)‘𝑦) = (𝐹𝑦))
114111, 113sylan9eqr 2798 . . . . . . . . 9 (((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦𝐴) → (𝑔𝑦) = (𝐹𝑦))
11548ad2antrr 724 . . . . . . . . . . 11 (((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦𝐴) → 𝐹:𝐴1-1𝐵)
116 f1fn 6739 . . . . . . . . . . 11 (𝐹:𝐴1-1𝐵𝐹 Fn 𝐴)
117115, 116syl 17 . . . . . . . . . 10 (((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦𝐴) → 𝐹 Fn 𝐴)
11825, 52fnsn 6559 . . . . . . . . . . 11 {⟨𝑧, 𝑥⟩} Fn {𝑧}
119118a1i 11 . . . . . . . . . 10 (((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦𝐴) → {⟨𝑧, 𝑥⟩} Fn {𝑧})
12065ad2antrr 724 . . . . . . . . . 10 (((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦𝐴) → (𝐴 ∩ {𝑧}) = ∅)
121 simpr 485 . . . . . . . . . 10 (((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦𝐴) → 𝑦𝐴)
122117, 119, 120, 121fvun1d 6934 . . . . . . . . 9 (((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦𝐴) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) = (𝐹𝑦))
123114, 122eqtr4d 2779 . . . . . . . 8 (((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦𝐴) → (𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦))
124123ralrimiva 3143 . . . . . . 7 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → ∀𝑦𝐴 (𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦))
125124biantrurd 533 . . . . . 6 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (∀𝑦 ∈ {𝑧} (𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ↔ (∀𝑦𝐴 (𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ∧ ∀𝑦 ∈ {𝑧} (𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦))))
126 ralunb 4151 . . . . . 6 (∀𝑦 ∈ (𝐴 ∪ {𝑧})(𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ↔ (∀𝑦𝐴 (𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ∧ ∀𝑦 ∈ {𝑧} (𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦)))
127125, 126bitr4di 288 . . . . 5 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (∀𝑦 ∈ {𝑧} (𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ↔ ∀𝑦 ∈ (𝐴 ∪ {𝑧})(𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦)))
12850fdmd 6679 . . . . . . . . . . 11 (𝜑 → dom 𝐹 = 𝐴)
129128eleq2d 2823 . . . . . . . . . 10 (𝜑 → (𝑧 ∈ dom 𝐹𝑧𝐴))
13030, 129mtbird 324 . . . . . . . . 9 (𝜑 → ¬ 𝑧 ∈ dom 𝐹)
131130adantr 481 . . . . . . . 8 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → ¬ 𝑧 ∈ dom 𝐹)
132 fsnunfv 7133 . . . . . . . 8 ((𝑧 ∈ V ∧ 𝑥 ∈ V ∧ ¬ 𝑧 ∈ dom 𝐹) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧) = 𝑥)
13325, 52, 131, 132mp3an12i 1465 . . . . . . 7 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧) = 𝑥)
134133eqeq2d 2747 . . . . . 6 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → ((𝑔𝑧) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧) ↔ (𝑔𝑧) = 𝑥))
135 fveq2 6842 . . . . . . . 8 (𝑦 = 𝑧 → (𝑔𝑦) = (𝑔𝑧))
136 fveq2 6842 . . . . . . . 8 (𝑦 = 𝑧 → ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧))
137135, 136eqeq12d 2752 . . . . . . 7 (𝑦 = 𝑧 → ((𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ↔ (𝑔𝑧) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧)))
13825, 137ralsn 4642 . . . . . 6 (∀𝑦 ∈ {𝑧} (𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ↔ (𝑔𝑧) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧))
139 eqcom 2743 . . . . . 6 (𝑥 = (𝑔𝑧) ↔ (𝑔𝑧) = 𝑥)
140134, 138, 1393bitr4g 313 . . . . 5 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (∀𝑦 ∈ {𝑧} (𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ↔ 𝑥 = (𝑔𝑧)))
141109, 127, 1403bitr2d 306 . . . 4 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (𝑔 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↔ 𝑥 = (𝑔𝑧)))
142141ex 413 . . 3 (𝜑 → ((((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝑔 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↔ 𝑥 = (𝑔𝑧))))
143101, 142biimtrid 241 . 2 (𝜑 → ((𝑔 ∈ {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝑔 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↔ 𝑥 = (𝑔𝑧))))
14413, 15, 47, 100, 143en3d 8929 1 (𝜑 → {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ≈ (𝐵 ∖ ran 𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  {cab 2713  wral 3064  Vcvv 3445  cdif 3907  cun 3908  cin 3909  wss 3910  c0 4282  {csn 4586  cop 4592   class class class wbr 5105  dom cdm 5633  ran crn 5634  cres 5635  cima 5636   Fn wfn 6491  wf 6492  1-1wf1 6493  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357  m cmap 8765  cen 8880  Fincfn 8883  1c1 11052   + caddc 11054  cle 11190  chash 14230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1o 8412  df-map 8767  df-en 8884  df-fin 8887
This theorem is referenced by: (None)
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