Theorem f1cofveqaeqALT 40324

Description: Alternate proof of f1cofveqaeq 40323, 1 essential step shorter, but having more bytes (305 vs. 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 6014	. . . . 5 ⊢ (𝐺:𝐴–1-1→𝐵 → 𝐺:𝐴⟶𝐵)
2		fvco3 6185	. . . . . . . 8 ⊢ ((𝐺:𝐴⟶𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋)))
3	2	adantrr 749	. . . . . . 7 ⊢ ((𝐺:𝐴⟶𝐵 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋)))
4		fvco3 6185	. . . . . . . 8 ⊢ ((𝐺:𝐴⟶𝐵 ∧ 𝑌 ∈ 𝐴) → ((𝐹 ∘ 𝐺)‘𝑌) = (𝐹‘(𝐺‘𝑌)))
5	4	adantrl 748	. . . . . . 7 ⊢ ((𝐺:𝐴⟶𝐵 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → ((𝐹 ∘ 𝐺)‘𝑌) = (𝐹‘(𝐺‘𝑌)))
6	3, 5	eqeq12d 2625	. . . . . 6 ⊢ ((𝐺:𝐴⟶𝐵 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → (((𝐹 ∘ 𝐺)‘𝑋) = ((𝐹 ∘ 𝐺)‘𝑌) ↔ (𝐹‘(𝐺‘𝑋)) = (𝐹‘(𝐺‘𝑌))))
7	6	ex 449	. . . . 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 444	. 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 6023	. . 3 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 ∘ 𝐺):𝐴–1-1→𝐶)
12		f1veqaeq 6418	. . 3 ⊢ (((𝐹 ∘ 𝐺):𝐴–1-1→𝐶 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → (((𝐹 ∘ 𝐺)‘𝑋) = ((𝐹 ∘ 𝐺)‘𝑌) → 𝑋 = 𝑌))
13	11, 12	sylan 487	. 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 249	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 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∘ ccom 5042  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fv 5812

This theorem is referenced by: (None)