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

Theorem f12dfv 7004
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 7003 . 2 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦)))
2 f12dfv.a . . . . 5 𝐴 = {𝑋, 𝑌}
32raleqi 3394 . . . 4 (∀𝑥𝐴𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦) ↔ ∀𝑥 ∈ {𝑋, 𝑌}∀𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦))
4 sneq 4550 . . . . . . . . 9 (𝑥 = 𝑋 → {𝑥} = {𝑋})
54difeq2d 4075 . . . . . . . 8 (𝑥 = 𝑋 → (𝐴 ∖ {𝑥}) = (𝐴 ∖ {𝑋}))
6 fveq2 6643 . . . . . . . . 9 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
76neeq1d 3066 . . . . . . . 8 (𝑥 = 𝑋 → ((𝐹𝑥) ≠ (𝐹𝑦) ↔ (𝐹𝑋) ≠ (𝐹𝑦)))
85, 7raleqbidv 3386 . . . . . . 7 (𝑥 = 𝑋 → (∀𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦) ↔ ∀𝑦 ∈ (𝐴 ∖ {𝑋})(𝐹𝑋) ≠ (𝐹𝑦)))
9 sneq 4550 . . . . . . . . 9 (𝑥 = 𝑌 → {𝑥} = {𝑌})
109difeq2d 4075 . . . . . . . 8 (𝑥 = 𝑌 → (𝐴 ∖ {𝑥}) = (𝐴 ∖ {𝑌}))
11 fveq2 6643 . . . . . . . . 9 (𝑥 = 𝑌 → (𝐹𝑥) = (𝐹𝑌))
1211neeq1d 3066 . . . . . . . 8 (𝑥 = 𝑌 → ((𝐹𝑥) ≠ (𝐹𝑦) ↔ (𝐹𝑌) ≠ (𝐹𝑦)))
1310, 12raleqbidv 3386 . . . . . . 7 (𝑥 = 𝑌 → (∀𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦) ↔ ∀𝑦 ∈ (𝐴 ∖ {𝑌})(𝐹𝑌) ≠ (𝐹𝑦)))
148, 13ralprg 4605 . . . . . 6 ((𝑋𝑈𝑌𝑉) → (∀𝑥 ∈ {𝑋, 𝑌}∀𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦) ↔ (∀𝑦 ∈ (𝐴 ∖ {𝑋})(𝐹𝑋) ≠ (𝐹𝑦) ∧ ∀𝑦 ∈ (𝐴 ∖ {𝑌})(𝐹𝑌) ≠ (𝐹𝑦))))
1514adantr 484 . . . . 5 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑥 ∈ {𝑋, 𝑌}∀𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦) ↔ (∀𝑦 ∈ (𝐴 ∖ {𝑋})(𝐹𝑋) ≠ (𝐹𝑦) ∧ ∀𝑦 ∈ (𝐴 ∖ {𝑌})(𝐹𝑌) ≠ (𝐹𝑦))))
162difeq1i 4071 . . . . . . . . . . 11 (𝐴 ∖ {𝑋}) = ({𝑋, 𝑌} ∖ {𝑋})
17 difprsn1 4706 . . . . . . . . . . 11 (𝑋𝑌 → ({𝑋, 𝑌} ∖ {𝑋}) = {𝑌})
1816, 17syl5eq 2868 . . . . . . . . . 10 (𝑋𝑌 → (𝐴 ∖ {𝑋}) = {𝑌})
1918adantl 485 . . . . . . . . 9 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (𝐴 ∖ {𝑋}) = {𝑌})
2019raleqdv 3396 . . . . . . . 8 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑦 ∈ (𝐴 ∖ {𝑋})(𝐹𝑋) ≠ (𝐹𝑦) ↔ ∀𝑦 ∈ {𝑌} (𝐹𝑋) ≠ (𝐹𝑦)))
21 fveq2 6643 . . . . . . . . . . . 12 (𝑦 = 𝑌 → (𝐹𝑦) = (𝐹𝑌))
2221neeq2d 3067 . . . . . . . . . . 11 (𝑦 = 𝑌 → ((𝐹𝑋) ≠ (𝐹𝑦) ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
2322ralsng 4586 . . . . . . . . . 10 (𝑌𝑉 → (∀𝑦 ∈ {𝑌} (𝐹𝑋) ≠ (𝐹𝑦) ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
2423adantl 485 . . . . . . . . 9 ((𝑋𝑈𝑌𝑉) → (∀𝑦 ∈ {𝑌} (𝐹𝑋) ≠ (𝐹𝑦) ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
2524adantr 484 . . . . . . . 8 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑦 ∈ {𝑌} (𝐹𝑋) ≠ (𝐹𝑦) ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
2620, 25bitrd 282 . . . . . . 7 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑦 ∈ (𝐴 ∖ {𝑋})(𝐹𝑋) ≠ (𝐹𝑦) ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
272difeq1i 4071 . . . . . . . . . . 11 (𝐴 ∖ {𝑌}) = ({𝑋, 𝑌} ∖ {𝑌})
28 difprsn2 4707 . . . . . . . . . . 11 (𝑋𝑌 → ({𝑋, 𝑌} ∖ {𝑌}) = {𝑋})
2927, 28syl5eq 2868 . . . . . . . . . 10 (𝑋𝑌 → (𝐴 ∖ {𝑌}) = {𝑋})
3029adantl 485 . . . . . . . . 9 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (𝐴 ∖ {𝑌}) = {𝑋})
3130raleqdv 3396 . . . . . . . 8 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑦 ∈ (𝐴 ∖ {𝑌})(𝐹𝑌) ≠ (𝐹𝑦) ↔ ∀𝑦 ∈ {𝑋} (𝐹𝑌) ≠ (𝐹𝑦)))
32 fveq2 6643 . . . . . . . . . . . 12 (𝑦 = 𝑋 → (𝐹𝑦) = (𝐹𝑋))
3332neeq2d 3067 . . . . . . . . . . 11 (𝑦 = 𝑋 → ((𝐹𝑌) ≠ (𝐹𝑦) ↔ (𝐹𝑌) ≠ (𝐹𝑋)))
3433ralsng 4586 . . . . . . . . . 10 (𝑋𝑈 → (∀𝑦 ∈ {𝑋} (𝐹𝑌) ≠ (𝐹𝑦) ↔ (𝐹𝑌) ≠ (𝐹𝑋)))
3534adantr 484 . . . . . . . . 9 ((𝑋𝑈𝑌𝑉) → (∀𝑦 ∈ {𝑋} (𝐹𝑌) ≠ (𝐹𝑦) ↔ (𝐹𝑌) ≠ (𝐹𝑋)))
3635adantr 484 . . . . . . . 8 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑦 ∈ {𝑋} (𝐹𝑌) ≠ (𝐹𝑦) ↔ (𝐹𝑌) ≠ (𝐹𝑋)))
3731, 36bitrd 282 . . . . . . 7 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑦 ∈ (𝐴 ∖ {𝑌})(𝐹𝑌) ≠ (𝐹𝑦) ↔ (𝐹𝑌) ≠ (𝐹𝑋)))
3826, 37anbi12d 633 . . . . . 6 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → ((∀𝑦 ∈ (𝐴 ∖ {𝑋})(𝐹𝑋) ≠ (𝐹𝑦) ∧ ∀𝑦 ∈ (𝐴 ∖ {𝑌})(𝐹𝑌) ≠ (𝐹𝑦)) ↔ ((𝐹𝑋) ≠ (𝐹𝑌) ∧ (𝐹𝑌) ≠ (𝐹𝑋))))
39 necom 3060 . . . . . . . 8 ((𝐹𝑋) ≠ (𝐹𝑌) ↔ (𝐹𝑌) ≠ (𝐹𝑋))
4039biimpi 219 . . . . . . 7 ((𝐹𝑋) ≠ (𝐹𝑌) → (𝐹𝑌) ≠ (𝐹𝑋))
4140pm4.71i 563 . . . . . 6 ((𝐹𝑋) ≠ (𝐹𝑌) ↔ ((𝐹𝑋) ≠ (𝐹𝑌) ∧ (𝐹𝑌) ≠ (𝐹𝑋)))
4238, 41syl6bbr 292 . . . . 5 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → ((∀𝑦 ∈ (𝐴 ∖ {𝑋})(𝐹𝑋) ≠ (𝐹𝑦) ∧ ∀𝑦 ∈ (𝐴 ∖ {𝑌})(𝐹𝑌) ≠ (𝐹𝑦)) ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
4315, 42bitrd 282 . . . 4 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑥 ∈ {𝑋, 𝑌}∀𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦) ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
443, 43syl5bb 286 . . 3 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑥𝐴𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦) ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
4544anbi2d 631 . 2 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → ((𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦)) ↔ (𝐹:𝐴𝐵 ∧ (𝐹𝑋) ≠ (𝐹𝑌))))
461, 45syl5bb 286 1 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ (𝐹𝑋) ≠ (𝐹𝑌))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wne 3007  wral 3126  cdif 3907  {csn 4540  {cpr 4542  wf 6324  1-1wf1 6325  cfv 6328
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-nel 3112  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fv 6336
This theorem is referenced by:  usgr2trlncl  27527
  Copyright terms: Public domain W3C validator