Theorem f1cofveqaeqALT 7011

Description: Alternate proof of f1cofveqaeq 7010, 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.)

Assertion

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))

Proof of Theorem f1cofveqaeqALT

Step	Hyp	Ref	Expression
1		f1f 6569	. . . . 5 ⊢ (𝐺:𝐴–1-1→𝐵 → 𝐺:𝐴⟶𝐵)
2		fvco3 6754	. . . . . . . 8 ⊢ ((𝐺:𝐴⟶𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋)))
3	2	adantrr 715	. . . . . . 7 ⊢ ((𝐺:𝐴⟶𝐵 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋)))
4		fvco3 6754	. . . . . . . 8 ⊢ ((𝐺:𝐴⟶𝐵 ∧ 𝑌 ∈ 𝐴) → ((𝐹 ∘ 𝐺)‘𝑌) = (𝐹‘(𝐺‘𝑌)))
5	4	adantrl 714	. . . . . . 7 ⊢ ((𝐺:𝐴⟶𝐵 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → ((𝐹 ∘ 𝐺)‘𝑌) = (𝐹‘(𝐺‘𝑌)))
6	3, 5	eqeq12d 2837	. . . . . 6 ⊢ ((𝐺:𝐴⟶𝐵 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → (((𝐹 ∘ 𝐺)‘𝑋) = ((𝐹 ∘ 𝐺)‘𝑌) ↔ (𝐹‘(𝐺‘𝑋)) = (𝐹‘(𝐺‘𝑌))))
7	6	ex 415	. . . . 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 484	. . 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 409	. 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 6579	. . 3 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 ∘ 𝐺):𝐴–1-1→𝐶)
12		f1veqaeq 7009	. . 3 ⊢ (((𝐹 ∘ 𝐺):𝐴–1-1→𝐶 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → (((𝐹 ∘ 𝐺)‘𝑋) = ((𝐹 ∘ 𝐺)‘𝑌) → 𝑋 = 𝑌))
13	11, 12	sylan 582	. 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 262	1 ⊢ (((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → ((𝐹‘(𝐺‘𝑋)) = (𝐹‘(𝐺‘𝑌)) → 𝑋 = 𝑌))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110   ∘ ccom 5553  ⟶wf 6345  –1-1→wf1 6346  ‘cfv 6349

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fv 6357

This theorem is referenced by: (None)