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| Description: Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.) | 
| Ref | Expression | 
|---|---|
| f1eq3 | ⊢ (𝐴 = 𝐵 → (𝐹:𝐶–1-1→𝐴 ↔ 𝐹:𝐶–1-1→𝐵)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | feq3 6717 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐹:𝐶⟶𝐴 ↔ 𝐹:𝐶⟶𝐵)) | |
| 2 | 1 | anbi1d 631 | . 2 ⊢ (𝐴 = 𝐵 → ((𝐹:𝐶⟶𝐴 ∧ Fun ◡𝐹) ↔ (𝐹:𝐶⟶𝐵 ∧ Fun ◡𝐹))) | 
| 3 | df-f1 6565 | . 2 ⊢ (𝐹:𝐶–1-1→𝐴 ↔ (𝐹:𝐶⟶𝐴 ∧ Fun ◡𝐹)) | |
| 4 | df-f1 6565 | . 2 ⊢ (𝐹:𝐶–1-1→𝐵 ↔ (𝐹:𝐶⟶𝐵 ∧ Fun ◡𝐹)) | |
| 5 | 2, 3, 4 | 3bitr4g 314 | 1 ⊢ (𝐴 = 𝐵 → (𝐹:𝐶–1-1→𝐴 ↔ 𝐹:𝐶–1-1→𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ◡ccnv 5683 Fun wfun 6554 ⟶wf 6556 –1-1→wf1 6557 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 df-cleq 2728 df-ss 3967 df-f 6564 df-f1 6565 | 
| This theorem is referenced by: f1oeq3 6837 f1eq123d 6839 tposf12 8277 brdom2g 8997 brdomgOLD 8999 1sdom2dom 9284 pwfseq 10705 f1linds 21846 isusgrs 29174 usgrstrrepe 29253 usgrexilem 29458 tocycval 33129 diaf1oN 41133 f1cof1b 47094 | 
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