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Mirrors > Home > MPE Home > Th. List > f1cofveqaeqALT | Structured version Visualization version GIF version |
Description: Alternate proof of f1cofveqaeq 7278, 1 essential step shorter, but having more bytes (305 versus 282). (Contributed by AV, 3-Feb-2021.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
f1cofveqaeqALT | ⊢ (((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → ((𝐹‘(𝐺‘𝑋)) = (𝐹‘(𝐺‘𝑌)) → 𝑋 = 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1f 6805 | . . . . 5 ⊢ (𝐺:𝐴–1-1→𝐵 → 𝐺:𝐴⟶𝐵) | |
2 | fvco3 7008 | . . . . . . . 8 ⊢ ((𝐺:𝐴⟶𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋))) | |
3 | 2 | adantrr 717 | . . . . . . 7 ⊢ ((𝐺:𝐴⟶𝐵 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋))) |
4 | fvco3 7008 | . . . . . . . 8 ⊢ ((𝐺:𝐴⟶𝐵 ∧ 𝑌 ∈ 𝐴) → ((𝐹 ∘ 𝐺)‘𝑌) = (𝐹‘(𝐺‘𝑌))) | |
5 | 4 | adantrl 716 | . . . . . . 7 ⊢ ((𝐺:𝐴⟶𝐵 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → ((𝐹 ∘ 𝐺)‘𝑌) = (𝐹‘(𝐺‘𝑌))) |
6 | 3, 5 | eqeq12d 2751 | . . . . . 6 ⊢ ((𝐺:𝐴⟶𝐵 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → (((𝐹 ∘ 𝐺)‘𝑋) = ((𝐹 ∘ 𝐺)‘𝑌) ↔ (𝐹‘(𝐺‘𝑋)) = (𝐹‘(𝐺‘𝑌)))) |
7 | 6 | ex 412 | . . . . 5 ⊢ (𝐺:𝐴⟶𝐵 → ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (((𝐹 ∘ 𝐺)‘𝑋) = ((𝐹 ∘ 𝐺)‘𝑌) ↔ (𝐹‘(𝐺‘𝑋)) = (𝐹‘(𝐺‘𝑌))))) |
8 | 1, 7 | syl 17 | . . . 4 ⊢ (𝐺:𝐴–1-1→𝐵 → ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (((𝐹 ∘ 𝐺)‘𝑋) = ((𝐹 ∘ 𝐺)‘𝑌) ↔ (𝐹‘(𝐺‘𝑋)) = (𝐹‘(𝐺‘𝑌))))) |
9 | 8 | adantl 481 | . . 3 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (((𝐹 ∘ 𝐺)‘𝑋) = ((𝐹 ∘ 𝐺)‘𝑌) ↔ (𝐹‘(𝐺‘𝑋)) = (𝐹‘(𝐺‘𝑌))))) |
10 | 9 | imp 406 | . 2 ⊢ (((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → (((𝐹 ∘ 𝐺)‘𝑋) = ((𝐹 ∘ 𝐺)‘𝑌) ↔ (𝐹‘(𝐺‘𝑋)) = (𝐹‘(𝐺‘𝑌)))) |
11 | f1co 6816 | . . 3 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 ∘ 𝐺):𝐴–1-1→𝐶) | |
12 | f1veqaeq 7277 | . . 3 ⊢ (((𝐹 ∘ 𝐺):𝐴–1-1→𝐶 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → (((𝐹 ∘ 𝐺)‘𝑋) = ((𝐹 ∘ 𝐺)‘𝑌) → 𝑋 = 𝑌)) | |
13 | 11, 12 | sylan 580 | . 2 ⊢ (((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → (((𝐹 ∘ 𝐺)‘𝑋) = ((𝐹 ∘ 𝐺)‘𝑌) → 𝑋 = 𝑌)) |
14 | 10, 13 | sylbird 260 | 1 ⊢ (((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → ((𝐹‘(𝐺‘𝑋)) = (𝐹‘(𝐺‘𝑌)) → 𝑋 = 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∘ ccom 5693 ⟶wf 6559 –1-1→wf1 6560 ‘cfv 6563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fv 6571 |
This theorem is referenced by: (None) |
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