MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  f12dfv Structured version   Visualization version   GIF version

Theorem f12dfv 7021
Description: A one-to-one function with a domain with at least two different elements in terms of function values. (Contributed by Alexander van der Vekens, 2-Mar-2018.)
Hypothesis
Ref Expression
f12dfv.a 𝐴 = {𝑋, 𝑌}
Assertion
Ref Expression
f12dfv (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ (𝐹𝑋) ≠ (𝐹𝑌))))

Proof of Theorem f12dfv
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dff14b 7020 . 2 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦)))
2 f12dfv.a . . . . 5 𝐴 = {𝑋, 𝑌}
32raleqi 3411 . . . 4 (∀𝑥𝐴𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦) ↔ ∀𝑥 ∈ {𝑋, 𝑌}∀𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦))
4 sneq 4567 . . . . . . . . 9 (𝑥 = 𝑋 → {𝑥} = {𝑋})
54difeq2d 4096 . . . . . . . 8 (𝑥 = 𝑋 → (𝐴 ∖ {𝑥}) = (𝐴 ∖ {𝑋}))
6 fveq2 6663 . . . . . . . . 9 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
76neeq1d 3072 . . . . . . . 8 (𝑥 = 𝑋 → ((𝐹𝑥) ≠ (𝐹𝑦) ↔ (𝐹𝑋) ≠ (𝐹𝑦)))
85, 7raleqbidv 3399 . . . . . . 7 (𝑥 = 𝑋 → (∀𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦) ↔ ∀𝑦 ∈ (𝐴 ∖ {𝑋})(𝐹𝑋) ≠ (𝐹𝑦)))
9 sneq 4567 . . . . . . . . 9 (𝑥 = 𝑌 → {𝑥} = {𝑌})
109difeq2d 4096 . . . . . . . 8 (𝑥 = 𝑌 → (𝐴 ∖ {𝑥}) = (𝐴 ∖ {𝑌}))
11 fveq2 6663 . . . . . . . . 9 (𝑥 = 𝑌 → (𝐹𝑥) = (𝐹𝑌))
1211neeq1d 3072 . . . . . . . 8 (𝑥 = 𝑌 → ((𝐹𝑥) ≠ (𝐹𝑦) ↔ (𝐹𝑌) ≠ (𝐹𝑦)))
1310, 12raleqbidv 3399 . . . . . . 7 (𝑥 = 𝑌 → (∀𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦) ↔ ∀𝑦 ∈ (𝐴 ∖ {𝑌})(𝐹𝑌) ≠ (𝐹𝑦)))
148, 13ralprg 4624 . . . . . 6 ((𝑋𝑈𝑌𝑉) → (∀𝑥 ∈ {𝑋, 𝑌}∀𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦) ↔ (∀𝑦 ∈ (𝐴 ∖ {𝑋})(𝐹𝑋) ≠ (𝐹𝑦) ∧ ∀𝑦 ∈ (𝐴 ∖ {𝑌})(𝐹𝑌) ≠ (𝐹𝑦))))
1514adantr 481 . . . . 5 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑥 ∈ {𝑋, 𝑌}∀𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦) ↔ (∀𝑦 ∈ (𝐴 ∖ {𝑋})(𝐹𝑋) ≠ (𝐹𝑦) ∧ ∀𝑦 ∈ (𝐴 ∖ {𝑌})(𝐹𝑌) ≠ (𝐹𝑦))))
162difeq1i 4092 . . . . . . . . . . 11 (𝐴 ∖ {𝑋}) = ({𝑋, 𝑌} ∖ {𝑋})
17 difprsn1 4725 . . . . . . . . . . 11 (𝑋𝑌 → ({𝑋, 𝑌} ∖ {𝑋}) = {𝑌})
1816, 17syl5eq 2865 . . . . . . . . . 10 (𝑋𝑌 → (𝐴 ∖ {𝑋}) = {𝑌})
1918adantl 482 . . . . . . . . 9 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (𝐴 ∖ {𝑋}) = {𝑌})
2019raleqdv 3413 . . . . . . . 8 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑦 ∈ (𝐴 ∖ {𝑋})(𝐹𝑋) ≠ (𝐹𝑦) ↔ ∀𝑦 ∈ {𝑌} (𝐹𝑋) ≠ (𝐹𝑦)))
21 fveq2 6663 . . . . . . . . . . . 12 (𝑦 = 𝑌 → (𝐹𝑦) = (𝐹𝑌))
2221neeq2d 3073 . . . . . . . . . . 11 (𝑦 = 𝑌 → ((𝐹𝑋) ≠ (𝐹𝑦) ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
2322ralsng 4605 . . . . . . . . . 10 (𝑌𝑉 → (∀𝑦 ∈ {𝑌} (𝐹𝑋) ≠ (𝐹𝑦) ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
2423adantl 482 . . . . . . . . 9 ((𝑋𝑈𝑌𝑉) → (∀𝑦 ∈ {𝑌} (𝐹𝑋) ≠ (𝐹𝑦) ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
2524adantr 481 . . . . . . . 8 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑦 ∈ {𝑌} (𝐹𝑋) ≠ (𝐹𝑦) ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
2620, 25bitrd 280 . . . . . . 7 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑦 ∈ (𝐴 ∖ {𝑋})(𝐹𝑋) ≠ (𝐹𝑦) ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
272difeq1i 4092 . . . . . . . . . . 11 (𝐴 ∖ {𝑌}) = ({𝑋, 𝑌} ∖ {𝑌})
28 difprsn2 4726 . . . . . . . . . . 11 (𝑋𝑌 → ({𝑋, 𝑌} ∖ {𝑌}) = {𝑋})
2927, 28syl5eq 2865 . . . . . . . . . 10 (𝑋𝑌 → (𝐴 ∖ {𝑌}) = {𝑋})
3029adantl 482 . . . . . . . . 9 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (𝐴 ∖ {𝑌}) = {𝑋})
3130raleqdv 3413 . . . . . . . 8 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑦 ∈ (𝐴 ∖ {𝑌})(𝐹𝑌) ≠ (𝐹𝑦) ↔ ∀𝑦 ∈ {𝑋} (𝐹𝑌) ≠ (𝐹𝑦)))
32 fveq2 6663 . . . . . . . . . . . 12 (𝑦 = 𝑋 → (𝐹𝑦) = (𝐹𝑋))
3332neeq2d 3073 . . . . . . . . . . 11 (𝑦 = 𝑋 → ((𝐹𝑌) ≠ (𝐹𝑦) ↔ (𝐹𝑌) ≠ (𝐹𝑋)))
3433ralsng 4605 . . . . . . . . . 10 (𝑋𝑈 → (∀𝑦 ∈ {𝑋} (𝐹𝑌) ≠ (𝐹𝑦) ↔ (𝐹𝑌) ≠ (𝐹𝑋)))
3534adantr 481 . . . . . . . . 9 ((𝑋𝑈𝑌𝑉) → (∀𝑦 ∈ {𝑋} (𝐹𝑌) ≠ (𝐹𝑦) ↔ (𝐹𝑌) ≠ (𝐹𝑋)))
3635adantr 481 . . . . . . . 8 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑦 ∈ {𝑋} (𝐹𝑌) ≠ (𝐹𝑦) ↔ (𝐹𝑌) ≠ (𝐹𝑋)))
3731, 36bitrd 280 . . . . . . 7 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑦 ∈ (𝐴 ∖ {𝑌})(𝐹𝑌) ≠ (𝐹𝑦) ↔ (𝐹𝑌) ≠ (𝐹𝑋)))
3826, 37anbi12d 630 . . . . . 6 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → ((∀𝑦 ∈ (𝐴 ∖ {𝑋})(𝐹𝑋) ≠ (𝐹𝑦) ∧ ∀𝑦 ∈ (𝐴 ∖ {𝑌})(𝐹𝑌) ≠ (𝐹𝑦)) ↔ ((𝐹𝑋) ≠ (𝐹𝑌) ∧ (𝐹𝑌) ≠ (𝐹𝑋))))
39 necom 3066 . . . . . . . 8 ((𝐹𝑋) ≠ (𝐹𝑌) ↔ (𝐹𝑌) ≠ (𝐹𝑋))
4039biimpi 217 . . . . . . 7 ((𝐹𝑋) ≠ (𝐹𝑌) → (𝐹𝑌) ≠ (𝐹𝑋))
4140pm4.71i 560 . . . . . 6 ((𝐹𝑋) ≠ (𝐹𝑌) ↔ ((𝐹𝑋) ≠ (𝐹𝑌) ∧ (𝐹𝑌) ≠ (𝐹𝑋)))
4238, 41syl6bbr 290 . . . . 5 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → ((∀𝑦 ∈ (𝐴 ∖ {𝑋})(𝐹𝑋) ≠ (𝐹𝑦) ∧ ∀𝑦 ∈ (𝐴 ∖ {𝑌})(𝐹𝑌) ≠ (𝐹𝑦)) ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
4315, 42bitrd 280 . . . 4 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑥 ∈ {𝑋, 𝑌}∀𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦) ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
443, 43syl5bb 284 . . 3 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑥𝐴𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦) ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
4544anbi2d 628 . 2 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → ((𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦)) ↔ (𝐹:𝐴𝐵 ∧ (𝐹𝑋) ≠ (𝐹𝑌))))
461, 45syl5bb 284 1 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ (𝐹𝑋) ≠ (𝐹𝑌))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wne 3013  wral 3135  cdif 3930  {csn 4557  {cpr 4559  wf 6344  1-1wf1 6345  cfv 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fv 6356
This theorem is referenced by:  usgr2trlncl  27468
  Copyright terms: Public domain W3C validator