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Mirrors > Home > MPE Home > Th. List > f1eq3 | Structured version Visualization version GIF version |
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 6691 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐹:𝐶⟶𝐴 ↔ 𝐹:𝐶⟶𝐵)) | |
2 | 1 | anbi1d 629 | . 2 ⊢ (𝐴 = 𝐵 → ((𝐹:𝐶⟶𝐴 ∧ Fun ◡𝐹) ↔ (𝐹:𝐶⟶𝐵 ∧ Fun ◡𝐹))) |
3 | df-f1 6539 | . 2 ⊢ (𝐹:𝐶–1-1→𝐴 ↔ (𝐹:𝐶⟶𝐴 ∧ Fun ◡𝐹)) | |
4 | df-f1 6539 | . 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 205 ∧ wa 395 = wceq 1533 ◡ccnv 5666 Fun wfun 6528 ⟶wf 6530 –1-1→wf1 6531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-v 3468 df-in 3948 df-ss 3958 df-f 6538 df-f1 6539 |
This theorem is referenced by: f1oeq3 6814 f1eq123d 6816 tposf12 8232 brdom2g 8948 brdomgOLD 8950 1sdom2dom 9244 pwfseq 10656 f1linds 21690 isusgrs 28888 usgrstrrepe 28964 usgrexilem 29169 tocycval 32738 diaf1oN 40495 f1cof1b 46295 |
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