Theorem f1cofveqaeqALT 7214

Description: Alternate proof of f1cofveqaeq 7213, 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 6738	. . . . 5 ⊢ (𝐺:𝐴–1-1→𝐵 → 𝐺:𝐴⟶𝐵)
2		fvco3 6941	. . . . . . . 8 ⊢ ((𝐺:𝐴⟶𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋)))
3	2	adantrr 718	. . . . . . 7 ⊢ ((𝐺:𝐴⟶𝐵 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋)))
4		fvco3 6941	. . . . . . . 8 ⊢ ((𝐺:𝐴⟶𝐵 ∧ 𝑌 ∈ 𝐴) → ((𝐹 ∘ 𝐺)‘𝑌) = (𝐹‘(𝐺‘𝑌)))
5	4	adantrl 717	. . . . . . 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 6749	. . 3 ⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 ∘ 𝐺):𝐴–1-1→𝐶)
12		f1veqaeq 7212	. . 3 ⊢ (((𝐹 ∘ 𝐺):𝐴–1-1→𝐶 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → (((𝐹 ∘ 𝐺)‘𝑋) = ((𝐹 ∘ 𝐺)‘𝑌) → 𝑋 = 𝑌))
13	11, 12	sylan 581	. 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 1542   ∈ wcel 2114   ∘ ccom 5636  ⟶wf 6496  –1-1→wf1 6497  ‘cfv 6500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fv 6508

This theorem is referenced by: (None)