Theorem 2f1fvneq 7295

Description: If two one-to-one functions are applied on different arguments, also the values are different. (Contributed by Alexander van der Vekens, 25-Jan-2018.) (Proof shortened by AV, 30-Oct-2025.)

Assertion

Ref	Expression
2f1fvneq	⊢ (((𝐸:𝐷⟶1-1𝑅 ∧ 𝐹:𝐶⟶1-1𝐷) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ 𝐴 ≠ 𝐵) → (((𝐸‘(𝐹‘𝐴)) = 𝑋 ∧ (𝐸‘(𝐹‘𝐵)) = 𝑌) → 𝑋 ≠ 𝑌))

Proof of Theorem 2f1fvneq

Step	Hyp	Ref	Expression
1		simp1l 1216	. . 3 ⊢ (((𝐸:𝐷⟶1-1𝑅 ∧ 𝐹:𝐶⟶1-1𝐷) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ 𝐴 ≠ 𝐵) → 𝐸:𝐷⟶1-1𝑅)
2		f1f 6808	. . . . . . 7 ⊢ (𝐹:𝐶⟶1-1𝐷 → 𝐹:𝐶⟶𝐷)
3	2	adantl 487	. . . . . 6 ⊢ ((𝐸:𝐷⟶1-1𝑅 ∧ 𝐹:𝐶⟶1-1𝐷) → 𝐹:𝐶⟶𝐷)
4	3	adantr 486	. . . . 5 ⊢ (((𝐸:𝐷⟶1-1𝑅 ∧ 𝐹:𝐶⟶1-1𝐷) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)) → 𝐹:𝐶⟶𝐷)
5		simpl 488	. . . . . 6 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶)
6	5	adantl 487	. . . . 5 ⊢ (((𝐸:𝐷⟶1-1𝑅 ∧ 𝐹:𝐶⟶1-1𝐷) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)) → 𝐴 ∈ 𝐶)
7	4, 6	ffvelcdmd 7117	. . . 4 ⊢ (((𝐸:𝐷⟶1-1𝑅 ∧ 𝐹:𝐶⟶1-1𝐷) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)) → (𝐹‘𝐴) ∈ 𝐷)
8	7	3adant3 1150	. . 3 ⊢ (((𝐸:𝐷⟶1-1𝑅 ∧ 𝐹:𝐶⟶1-1𝐷) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ 𝐴 ≠ 𝐵) → (𝐹‘𝐴) ∈ 𝐷)
9		simpr 490	. . . . . 6 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → 𝐵 ∈ 𝐶)
10	9	adantl 487	. . . . 5 ⊢ (((𝐸:𝐷⟶1-1𝑅 ∧ 𝐹:𝐶⟶1-1𝐷) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)) → 𝐵 ∈ 𝐶)
11	4, 10	ffvelcdmd 7117	. . . 4 ⊢ (((𝐸:𝐷⟶1-1𝑅 ∧ 𝐹:𝐶⟶1-1𝐷) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)) → (𝐹‘𝐵) ∈ 𝐷)
12	11	3adant3 1150	. . 3 ⊢ (((𝐸:𝐷⟶1-1𝑅 ∧ 𝐹:𝐶⟶1-1𝐷) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ 𝐴 ≠ 𝐵) → (𝐹‘𝐵) ∈ 𝐷)
13		simpr 490	. . . 4 ⊢ ((𝐸:𝐷⟶1-1𝑅 ∧ 𝐹:𝐶⟶1-1𝐷) → 𝐹:𝐶⟶1-1𝐷)
14		df-3an 1105	. . . . 5 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ∧ 𝐴 ≠ 𝐵) ↔ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ 𝐴 ≠ 𝐵))
15	14	biimpri 231	. . . 4 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ 𝐴 ≠ 𝐵) → (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ∧ 𝐴 ≠ 𝐵))
16		dff14i 7294	. . . 4 ⊢ ((𝐹:𝐶⟶1-1𝐷 ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ∧ 𝐴 ≠ 𝐵)) → (𝐹‘𝐴) ≠ (𝐹‘𝐵))
17	13, 15, 16	syl3an132 1184	. . 3 ⊢ (((𝐸:𝐷⟶1-1𝑅 ∧ 𝐹:𝐶⟶1-1𝐷) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ 𝐴 ≠ 𝐵) → (𝐹‘𝐴) ≠ (𝐹‘𝐵))
18		dff14i 7294	. . 3 ⊢ ((𝐸:𝐷⟶1-1𝑅 ∧ ((𝐹‘𝐴) ∈ 𝐷 ∧ (𝐹‘𝐵) ∈ 𝐷 ∧ (𝐹‘𝐴) ≠ (𝐹‘𝐵))) → (𝐸‘(𝐹‘𝐴)) ≠ (𝐸‘(𝐹‘𝐵)))
19	1, 8, 12, 17, 18	syl13anc 1400	. 2 ⊢ (((𝐸:𝐷⟶1-1𝑅 ∧ 𝐹:𝐶⟶1-1𝐷) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ 𝐴 ≠ 𝐵) → (𝐸‘(𝐹‘𝐴)) ≠ (𝐸‘(𝐹‘𝐵)))
20		simpl 488	. . 3 ⊢ (((𝐸‘(𝐹‘𝐴)) = 𝑋 ∧ (𝐸‘(𝐹‘𝐵)) = 𝑌) → (𝐸‘(𝐹‘𝐴)) = 𝑋)
21		simpr 490	. . 3 ⊢ (((𝐸‘(𝐹‘𝐴)) = 𝑋 ∧ (𝐸‘(𝐹‘𝐵)) = 𝑌) → (𝐸‘(𝐹‘𝐵)) = 𝑌)
22	20, 21	neeq12d 3020	. 2 ⊢ (((𝐸‘(𝐹‘𝐴)) = 𝑋 ∧ (𝐸‘(𝐹‘𝐵)) = 𝑌) → ((𝐸‘(𝐹‘𝐴)) ≠ (𝐸‘(𝐹‘𝐵)) ↔ 𝑋 ≠ 𝑌))
23	19, 22	syl5ibcom 248	1 ⊢ (((𝐸:𝐷⟶1-1𝑅 ∧ 𝐹:𝐶⟶1-1𝐷) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ 𝐴 ≠ 𝐵) → (((𝐸‘(𝐹‘𝐴)) = 𝑋 ∧ (𝐸‘(𝐹‘𝐵)) = 𝑌) → 𝑋 ≠ 𝑌))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ⟶wf 6562  ⟶1-1wf1 6563  ‘cfv 6566

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fv 6574

This theorem is used by:  usgr2pthlem  30406