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Theorem f12dfv 6569
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 6568 . 2 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦)))
2 f12dfv.a . . . . 5 𝐴 = {𝑋, 𝑌}
32raleqi 3172 . . . 4 (∀𝑥𝐴𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦) ↔ ∀𝑥 ∈ {𝑋, 𝑌}∀𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦))
4 sneq 4220 . . . . . . . . 9 (𝑥 = 𝑋 → {𝑥} = {𝑋})
54difeq2d 3761 . . . . . . . 8 (𝑥 = 𝑋 → (𝐴 ∖ {𝑥}) = (𝐴 ∖ {𝑋}))
6 fveq2 6229 . . . . . . . . 9 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
76neeq1d 2882 . . . . . . . 8 (𝑥 = 𝑋 → ((𝐹𝑥) ≠ (𝐹𝑦) ↔ (𝐹𝑋) ≠ (𝐹𝑦)))
85, 7raleqbidv 3182 . . . . . . 7 (𝑥 = 𝑋 → (∀𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦) ↔ ∀𝑦 ∈ (𝐴 ∖ {𝑋})(𝐹𝑋) ≠ (𝐹𝑦)))
9 sneq 4220 . . . . . . . . 9 (𝑥 = 𝑌 → {𝑥} = {𝑌})
109difeq2d 3761 . . . . . . . 8 (𝑥 = 𝑌 → (𝐴 ∖ {𝑥}) = (𝐴 ∖ {𝑌}))
11 fveq2 6229 . . . . . . . . 9 (𝑥 = 𝑌 → (𝐹𝑥) = (𝐹𝑌))
1211neeq1d 2882 . . . . . . . 8 (𝑥 = 𝑌 → ((𝐹𝑥) ≠ (𝐹𝑦) ↔ (𝐹𝑌) ≠ (𝐹𝑦)))
1310, 12raleqbidv 3182 . . . . . . 7 (𝑥 = 𝑌 → (∀𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦) ↔ ∀𝑦 ∈ (𝐴 ∖ {𝑌})(𝐹𝑌) ≠ (𝐹𝑦)))
148, 13ralprg 4266 . . . . . 6 ((𝑋𝑈𝑌𝑉) → (∀𝑥 ∈ {𝑋, 𝑌}∀𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦) ↔ (∀𝑦 ∈ (𝐴 ∖ {𝑋})(𝐹𝑋) ≠ (𝐹𝑦) ∧ ∀𝑦 ∈ (𝐴 ∖ {𝑌})(𝐹𝑌) ≠ (𝐹𝑦))))
1514adantr 480 . . . . 5 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑥 ∈ {𝑋, 𝑌}∀𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦) ↔ (∀𝑦 ∈ (𝐴 ∖ {𝑋})(𝐹𝑋) ≠ (𝐹𝑦) ∧ ∀𝑦 ∈ (𝐴 ∖ {𝑌})(𝐹𝑌) ≠ (𝐹𝑦))))
162difeq1i 3757 . . . . . . . . . . 11 (𝐴 ∖ {𝑋}) = ({𝑋, 𝑌} ∖ {𝑋})
17 difprsn1 4362 . . . . . . . . . . 11 (𝑋𝑌 → ({𝑋, 𝑌} ∖ {𝑋}) = {𝑌})
1816, 17syl5eq 2697 . . . . . . . . . 10 (𝑋𝑌 → (𝐴 ∖ {𝑋}) = {𝑌})
1918adantl 481 . . . . . . . . 9 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (𝐴 ∖ {𝑋}) = {𝑌})
2019raleqdv 3174 . . . . . . . 8 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑦 ∈ (𝐴 ∖ {𝑋})(𝐹𝑋) ≠ (𝐹𝑦) ↔ ∀𝑦 ∈ {𝑌} (𝐹𝑋) ≠ (𝐹𝑦)))
21 fveq2 6229 . . . . . . . . . . . 12 (𝑦 = 𝑌 → (𝐹𝑦) = (𝐹𝑌))
2221neeq2d 2883 . . . . . . . . . . 11 (𝑦 = 𝑌 → ((𝐹𝑋) ≠ (𝐹𝑦) ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
2322ralsng 4250 . . . . . . . . . 10 (𝑌𝑉 → (∀𝑦 ∈ {𝑌} (𝐹𝑋) ≠ (𝐹𝑦) ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
2423adantl 481 . . . . . . . . 9 ((𝑋𝑈𝑌𝑉) → (∀𝑦 ∈ {𝑌} (𝐹𝑋) ≠ (𝐹𝑦) ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
2524adantr 480 . . . . . . . 8 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑦 ∈ {𝑌} (𝐹𝑋) ≠ (𝐹𝑦) ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
2620, 25bitrd 268 . . . . . . 7 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑦 ∈ (𝐴 ∖ {𝑋})(𝐹𝑋) ≠ (𝐹𝑦) ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
272difeq1i 3757 . . . . . . . . . . 11 (𝐴 ∖ {𝑌}) = ({𝑋, 𝑌} ∖ {𝑌})
28 difprsn2 4363 . . . . . . . . . . 11 (𝑋𝑌 → ({𝑋, 𝑌} ∖ {𝑌}) = {𝑋})
2927, 28syl5eq 2697 . . . . . . . . . 10 (𝑋𝑌 → (𝐴 ∖ {𝑌}) = {𝑋})
3029adantl 481 . . . . . . . . 9 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (𝐴 ∖ {𝑌}) = {𝑋})
3130raleqdv 3174 . . . . . . . 8 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑦 ∈ (𝐴 ∖ {𝑌})(𝐹𝑌) ≠ (𝐹𝑦) ↔ ∀𝑦 ∈ {𝑋} (𝐹𝑌) ≠ (𝐹𝑦)))
32 fveq2 6229 . . . . . . . . . . . 12 (𝑦 = 𝑋 → (𝐹𝑦) = (𝐹𝑋))
3332neeq2d 2883 . . . . . . . . . . 11 (𝑦 = 𝑋 → ((𝐹𝑌) ≠ (𝐹𝑦) ↔ (𝐹𝑌) ≠ (𝐹𝑋)))
3433ralsng 4250 . . . . . . . . . 10 (𝑋𝑈 → (∀𝑦 ∈ {𝑋} (𝐹𝑌) ≠ (𝐹𝑦) ↔ (𝐹𝑌) ≠ (𝐹𝑋)))
3534adantr 480 . . . . . . . . 9 ((𝑋𝑈𝑌𝑉) → (∀𝑦 ∈ {𝑋} (𝐹𝑌) ≠ (𝐹𝑦) ↔ (𝐹𝑌) ≠ (𝐹𝑋)))
3635adantr 480 . . . . . . . 8 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑦 ∈ {𝑋} (𝐹𝑌) ≠ (𝐹𝑦) ↔ (𝐹𝑌) ≠ (𝐹𝑋)))
3731, 36bitrd 268 . . . . . . 7 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑦 ∈ (𝐴 ∖ {𝑌})(𝐹𝑌) ≠ (𝐹𝑦) ↔ (𝐹𝑌) ≠ (𝐹𝑋)))
3826, 37anbi12d 747 . . . . . 6 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → ((∀𝑦 ∈ (𝐴 ∖ {𝑋})(𝐹𝑋) ≠ (𝐹𝑦) ∧ ∀𝑦 ∈ (𝐴 ∖ {𝑌})(𝐹𝑌) ≠ (𝐹𝑦)) ↔ ((𝐹𝑋) ≠ (𝐹𝑌) ∧ (𝐹𝑌) ≠ (𝐹𝑋))))
39 necom 2876 . . . . . . . 8 ((𝐹𝑋) ≠ (𝐹𝑌) ↔ (𝐹𝑌) ≠ (𝐹𝑋))
4039biimpi 206 . . . . . . 7 ((𝐹𝑋) ≠ (𝐹𝑌) → (𝐹𝑌) ≠ (𝐹𝑋))
4140pm4.71i 665 . . . . . 6 ((𝐹𝑋) ≠ (𝐹𝑌) ↔ ((𝐹𝑋) ≠ (𝐹𝑌) ∧ (𝐹𝑌) ≠ (𝐹𝑋)))
4238, 41syl6bbr 278 . . . . 5 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → ((∀𝑦 ∈ (𝐴 ∖ {𝑋})(𝐹𝑋) ≠ (𝐹𝑦) ∧ ∀𝑦 ∈ (𝐴 ∖ {𝑌})(𝐹𝑌) ≠ (𝐹𝑦)) ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
4315, 42bitrd 268 . . . 4 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑥 ∈ {𝑋, 𝑌}∀𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦) ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
443, 43syl5bb 272 . . 3 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑥𝐴𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦) ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
4544anbi2d 740 . 2 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → ((𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦)) ↔ (𝐹:𝐴𝐵 ∧ (𝐹𝑋) ≠ (𝐹𝑌))))
461, 45syl5bb 272 1 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ (𝐹𝑋) ≠ (𝐹𝑌))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wne 2823  wral 2941  cdif 3604  {csn 4210  {cpr 4212  wf 5922  1-1wf1 5923  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fv 5934
This theorem is referenced by:  usgr2trlncl  26712
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