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Mirrors > Home > MPE Home > Th. List > f1imaeq | Structured version Visualization version GIF version |
Description: Taking images under a one-to-one function preserves equality. (Contributed by Stefan O'Rear, 30-Oct-2014.) |
Ref | Expression |
---|---|
f1imaeq | ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) → ((𝐹 “ 𝐶) = (𝐹 “ 𝐷) ↔ 𝐶 = 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1imass 7212 | . . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) → ((𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷) ↔ 𝐶 ⊆ 𝐷)) | |
2 | f1imass 7212 | . . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐷 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴)) → ((𝐹 “ 𝐷) ⊆ (𝐹 “ 𝐶) ↔ 𝐷 ⊆ 𝐶)) | |
3 | 2 | ancom2s 649 | . . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) → ((𝐹 “ 𝐷) ⊆ (𝐹 “ 𝐶) ↔ 𝐷 ⊆ 𝐶)) |
4 | 1, 3 | anbi12d 632 | . 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) → (((𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷) ∧ (𝐹 “ 𝐷) ⊆ (𝐹 “ 𝐶)) ↔ (𝐶 ⊆ 𝐷 ∧ 𝐷 ⊆ 𝐶))) |
5 | eqss 3960 | . 2 ⊢ ((𝐹 “ 𝐶) = (𝐹 “ 𝐷) ↔ ((𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷) ∧ (𝐹 “ 𝐷) ⊆ (𝐹 “ 𝐶))) | |
6 | eqss 3960 | . 2 ⊢ (𝐶 = 𝐷 ↔ (𝐶 ⊆ 𝐷 ∧ 𝐷 ⊆ 𝐶)) | |
7 | 4, 5, 6 | 3bitr4g 314 | 1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) → ((𝐹 “ 𝐶) = (𝐹 “ 𝐷) ↔ 𝐶 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ⊆ wss 3911 “ cima 5637 –1-1→wf1 6494 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fv 6505 |
This theorem is referenced by: f1imapss 7214 dfac12lem2 10085 hmeoimaf1o 23137 imasf1oxms 23861 isomuspgrlem2c 46108 |
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