f1oeq3 - Metamath Proof Explorer

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Theorem f1oeq3 6606

Description: Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)

**Assertion**
Ref	Expression
f1oeq3	⊢ (𝐴 = 𝐵 → (𝐹:𝐶–1-1-onto→𝐴 ↔ 𝐹:𝐶–1-1-onto→𝐵))

**Proof of Theorem f1oeq3**
Step	Hyp	Ref	Expression
1		f1eq3 6572	. . 3 ⊢ (𝐴 = 𝐵 → (𝐹:𝐶–1-1→𝐴 ↔ 𝐹:𝐶–1-1→𝐵))
2		foeq3 6588	. . 3 ⊢ (𝐴 = 𝐵 → (𝐹:𝐶–onto→𝐴 ↔ 𝐹:𝐶–onto→𝐵))
3	1, 2	anbi12d 632	. 2 ⊢ (𝐴 = 𝐵 → ((𝐹:𝐶–1-1→𝐴 ∧ 𝐹:𝐶–onto→𝐴) ↔ (𝐹:𝐶–1-1→𝐵 ∧ 𝐹:𝐶–onto→𝐵)))
4		df-f1o 6362	. 2 ⊢ (𝐹:𝐶–1-1-onto→𝐴 ↔ (𝐹:𝐶–1-1→𝐴 ∧ 𝐹:𝐶–onto→𝐴))
5		df-f1o 6362	. 2 ⊢ (𝐹:𝐶–1-1-onto→𝐵 ↔ (𝐹:𝐶–1-1→𝐵 ∧ 𝐹:𝐶–onto→𝐵))
6	3, 4, 5	3bitr4g 316	1 ⊢ (𝐴 = 𝐵 → (𝐹:𝐶–1-1-onto→𝐴 ↔ 𝐹:𝐶–1-1-onto→𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 –1-1→wf1 6352 –onto→wfo 6353 –1-1-onto→wf1o 6354

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-in 3943 df-ss 3952 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362

This theorem is referenced by: f1oeq23 6607 f1oeq123d 6610 f1oeq3d 6612 f1ores 6629 resin 6636 isoeq5 7074 bren 8518 xpcomf1o 8606 isinf 8731 cnfcom2 9165 fin1a2lem6 9827 pwfseqlem5 10085 pwfseq 10086 hashgf1o 13340 axdc4uzlem 13352 sumeq1 15045 prodeq1f 15262 unbenlem 16244 4sqlem11 16291 gsumvalx 17886 cayley 18542 cayleyth 18543 ovolicc2lem4 24121 logf1o2 25233 uspgrf1oedg 26958 wlkiswwlks2lem4 27650 clwwlknonclwlknonf1o 28141 dlwwlknondlwlknonf1o 28144 adjbd1o 29862 rinvf1o 30375 cshf1o 30636 eulerpartgbij 31630 eulerpartlemgh 31636 derangval 32414 subfacp1lem2a 32427 subfacp1lem3 32429 subfacp1lem5 32431 mrsubff1o 32762 msubff1o 32804 bj-finsumval0 34570 f1omptsnlem 34620 f1omptsn 34621 poimirlem4 34911 poimirlem9 34916 poimirlem15 34922 ismtyval 35093 ismrer1 35131 lautset 37233 pautsetN 37249 hvmap1o2 38916 pwfi2f1o 39716 imasgim 39720 alephiso2 39937 isomushgr 44011 uspgrsprfo 44043