Theorem f1cofveqaeqALT 7279

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.)

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 6804	. . . . 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 2753	. . . . . 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 6815	. . 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))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ∘ ccom 5689  ⟶wf 6557  –1-1→wf1 6558  ‘cfv 6561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fv 6569

This theorem is referenced by: (None)