MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1cofveqaeq Structured version   Visualization version   GIF version

Theorem f1cofveqaeq 6678
Description: If the values of a composition of one-to-one functions for two arguments are equal, the arguments themselves must be equal. (Contributed by AV, 3-Feb-2021.)
Assertion
Ref Expression
f1cofveqaeq (((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) ∧ (𝑋𝐴𝑌𝐴)) → ((𝐹‘(𝐺𝑋)) = (𝐹‘(𝐺𝑌)) → 𝑋 = 𝑌))

Proof of Theorem f1cofveqaeq
StepHypRef Expression
1 simpl 474 . . 3 ((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) → 𝐹:𝐵1-1𝐶)
2 f1f 6262 . . . . . 6 (𝐺:𝐴1-1𝐵𝐺:𝐴𝐵)
3 ffvelrn 6520 . . . . . . . 8 ((𝐺:𝐴𝐵𝑋𝐴) → (𝐺𝑋) ∈ 𝐵)
43ex 449 . . . . . . 7 (𝐺:𝐴𝐵 → (𝑋𝐴 → (𝐺𝑋) ∈ 𝐵))
5 ffvelrn 6520 . . . . . . . 8 ((𝐺:𝐴𝐵𝑌𝐴) → (𝐺𝑌) ∈ 𝐵)
65ex 449 . . . . . . 7 (𝐺:𝐴𝐵 → (𝑌𝐴 → (𝐺𝑌) ∈ 𝐵))
74, 6anim12d 587 . . . . . 6 (𝐺:𝐴𝐵 → ((𝑋𝐴𝑌𝐴) → ((𝐺𝑋) ∈ 𝐵 ∧ (𝐺𝑌) ∈ 𝐵)))
82, 7syl 17 . . . . 5 (𝐺:𝐴1-1𝐵 → ((𝑋𝐴𝑌𝐴) → ((𝐺𝑋) ∈ 𝐵 ∧ (𝐺𝑌) ∈ 𝐵)))
98adantl 473 . . . 4 ((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) → ((𝑋𝐴𝑌𝐴) → ((𝐺𝑋) ∈ 𝐵 ∧ (𝐺𝑌) ∈ 𝐵)))
109imp 444 . . 3 (((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) ∧ (𝑋𝐴𝑌𝐴)) → ((𝐺𝑋) ∈ 𝐵 ∧ (𝐺𝑌) ∈ 𝐵))
11 f1veqaeq 6677 . . 3 ((𝐹:𝐵1-1𝐶 ∧ ((𝐺𝑋) ∈ 𝐵 ∧ (𝐺𝑌) ∈ 𝐵)) → ((𝐹‘(𝐺𝑋)) = (𝐹‘(𝐺𝑌)) → (𝐺𝑋) = (𝐺𝑌)))
121, 10, 11syl2an2r 911 . 2 (((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) ∧ (𝑋𝐴𝑌𝐴)) → ((𝐹‘(𝐺𝑋)) = (𝐹‘(𝐺𝑌)) → (𝐺𝑋) = (𝐺𝑌)))
13 f1veqaeq 6677 . . 3 ((𝐺:𝐴1-1𝐵 ∧ (𝑋𝐴𝑌𝐴)) → ((𝐺𝑋) = (𝐺𝑌) → 𝑋 = 𝑌))
1413adantll 752 . 2 (((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) ∧ (𝑋𝐴𝑌𝐴)) → ((𝐺𝑋) = (𝐺𝑌) → 𝑋 = 𝑌))
1512, 14syld 47 1 (((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) ∧ (𝑋𝐴𝑌𝐴)) → ((𝐹‘(𝐺𝑋)) = (𝐹‘(𝐺𝑌)) → 𝑋 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wf 6045  1-1wf1 6046  cfv 6049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fv 6057
This theorem is referenced by:  uspgrn2crct  26911
  Copyright terms: Public domain W3C validator