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Theorem f12dfv 7282
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 7281 . 2 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦)))
2 f12dfv.a . . . . 5 𝐴 = {𝑋, 𝑌}
32raleqi 3312 . . . 4 (∀𝑥𝐴𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦) ↔ ∀𝑥 ∈ {𝑋, 𝑌}∀𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦))
4 sneq 4640 . . . . . . . . 9 (𝑥 = 𝑋 → {𝑥} = {𝑋})
54difeq2d 4118 . . . . . . . 8 (𝑥 = 𝑋 → (𝐴 ∖ {𝑥}) = (𝐴 ∖ {𝑋}))
6 fveq2 6896 . . . . . . . . 9 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
76neeq1d 2989 . . . . . . . 8 (𝑥 = 𝑋 → ((𝐹𝑥) ≠ (𝐹𝑦) ↔ (𝐹𝑋) ≠ (𝐹𝑦)))
85, 7raleqbidv 3329 . . . . . . 7 (𝑥 = 𝑋 → (∀𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦) ↔ ∀𝑦 ∈ (𝐴 ∖ {𝑋})(𝐹𝑋) ≠ (𝐹𝑦)))
9 sneq 4640 . . . . . . . . 9 (𝑥 = 𝑌 → {𝑥} = {𝑌})
109difeq2d 4118 . . . . . . . 8 (𝑥 = 𝑌 → (𝐴 ∖ {𝑥}) = (𝐴 ∖ {𝑌}))
11 fveq2 6896 . . . . . . . . 9 (𝑥 = 𝑌 → (𝐹𝑥) = (𝐹𝑌))
1211neeq1d 2989 . . . . . . . 8 (𝑥 = 𝑌 → ((𝐹𝑥) ≠ (𝐹𝑦) ↔ (𝐹𝑌) ≠ (𝐹𝑦)))
1310, 12raleqbidv 3329 . . . . . . 7 (𝑥 = 𝑌 → (∀𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦) ↔ ∀𝑦 ∈ (𝐴 ∖ {𝑌})(𝐹𝑌) ≠ (𝐹𝑦)))
148, 13ralprg 4700 . . . . . 6 ((𝑋𝑈𝑌𝑉) → (∀𝑥 ∈ {𝑋, 𝑌}∀𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦) ↔ (∀𝑦 ∈ (𝐴 ∖ {𝑋})(𝐹𝑋) ≠ (𝐹𝑦) ∧ ∀𝑦 ∈ (𝐴 ∖ {𝑌})(𝐹𝑌) ≠ (𝐹𝑦))))
1514adantr 479 . . . . 5 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑥 ∈ {𝑋, 𝑌}∀𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦) ↔ (∀𝑦 ∈ (𝐴 ∖ {𝑋})(𝐹𝑋) ≠ (𝐹𝑦) ∧ ∀𝑦 ∈ (𝐴 ∖ {𝑌})(𝐹𝑌) ≠ (𝐹𝑦))))
162difeq1i 4114 . . . . . . . . . . 11 (𝐴 ∖ {𝑋}) = ({𝑋, 𝑌} ∖ {𝑋})
17 difprsn1 4805 . . . . . . . . . . 11 (𝑋𝑌 → ({𝑋, 𝑌} ∖ {𝑋}) = {𝑌})
1816, 17eqtrid 2777 . . . . . . . . . 10 (𝑋𝑌 → (𝐴 ∖ {𝑋}) = {𝑌})
1918adantl 480 . . . . . . . . 9 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (𝐴 ∖ {𝑋}) = {𝑌})
2019raleqdv 3314 . . . . . . . 8 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑦 ∈ (𝐴 ∖ {𝑋})(𝐹𝑋) ≠ (𝐹𝑦) ↔ ∀𝑦 ∈ {𝑌} (𝐹𝑋) ≠ (𝐹𝑦)))
21 fveq2 6896 . . . . . . . . . . . 12 (𝑦 = 𝑌 → (𝐹𝑦) = (𝐹𝑌))
2221neeq2d 2990 . . . . . . . . . . 11 (𝑦 = 𝑌 → ((𝐹𝑋) ≠ (𝐹𝑦) ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
2322ralsng 4679 . . . . . . . . . 10 (𝑌𝑉 → (∀𝑦 ∈ {𝑌} (𝐹𝑋) ≠ (𝐹𝑦) ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
2423adantl 480 . . . . . . . . 9 ((𝑋𝑈𝑌𝑉) → (∀𝑦 ∈ {𝑌} (𝐹𝑋) ≠ (𝐹𝑦) ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
2524adantr 479 . . . . . . . 8 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑦 ∈ {𝑌} (𝐹𝑋) ≠ (𝐹𝑦) ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
2620, 25bitrd 278 . . . . . . 7 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑦 ∈ (𝐴 ∖ {𝑋})(𝐹𝑋) ≠ (𝐹𝑦) ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
272difeq1i 4114 . . . . . . . . . . 11 (𝐴 ∖ {𝑌}) = ({𝑋, 𝑌} ∖ {𝑌})
28 difprsn2 4806 . . . . . . . . . . 11 (𝑋𝑌 → ({𝑋, 𝑌} ∖ {𝑌}) = {𝑋})
2927, 28eqtrid 2777 . . . . . . . . . 10 (𝑋𝑌 → (𝐴 ∖ {𝑌}) = {𝑋})
3029adantl 480 . . . . . . . . 9 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (𝐴 ∖ {𝑌}) = {𝑋})
3130raleqdv 3314 . . . . . . . 8 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑦 ∈ (𝐴 ∖ {𝑌})(𝐹𝑌) ≠ (𝐹𝑦) ↔ ∀𝑦 ∈ {𝑋} (𝐹𝑌) ≠ (𝐹𝑦)))
32 fveq2 6896 . . . . . . . . . . . 12 (𝑦 = 𝑋 → (𝐹𝑦) = (𝐹𝑋))
3332neeq2d 2990 . . . . . . . . . . 11 (𝑦 = 𝑋 → ((𝐹𝑌) ≠ (𝐹𝑦) ↔ (𝐹𝑌) ≠ (𝐹𝑋)))
3433ralsng 4679 . . . . . . . . . 10 (𝑋𝑈 → (∀𝑦 ∈ {𝑋} (𝐹𝑌) ≠ (𝐹𝑦) ↔ (𝐹𝑌) ≠ (𝐹𝑋)))
3534adantr 479 . . . . . . . . 9 ((𝑋𝑈𝑌𝑉) → (∀𝑦 ∈ {𝑋} (𝐹𝑌) ≠ (𝐹𝑦) ↔ (𝐹𝑌) ≠ (𝐹𝑋)))
3635adantr 479 . . . . . . . 8 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑦 ∈ {𝑋} (𝐹𝑌) ≠ (𝐹𝑦) ↔ (𝐹𝑌) ≠ (𝐹𝑋)))
3731, 36bitrd 278 . . . . . . 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 2983 . . . . . . . 8 ((𝐹𝑋) ≠ (𝐹𝑌) ↔ (𝐹𝑌) ≠ (𝐹𝑋))
4039biimpi 215 . . . . . . 7 ((𝐹𝑋) ≠ (𝐹𝑌) → (𝐹𝑌) ≠ (𝐹𝑋))
4140pm4.71i 558 . . . . . 6 ((𝐹𝑋) ≠ (𝐹𝑌) ↔ ((𝐹𝑋) ≠ (𝐹𝑌) ∧ (𝐹𝑌) ≠ (𝐹𝑋)))
4238, 41bitr4di 288 . . . . 5 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → ((∀𝑦 ∈ (𝐴 ∖ {𝑋})(𝐹𝑋) ≠ (𝐹𝑦) ∧ ∀𝑦 ∈ (𝐴 ∖ {𝑌})(𝐹𝑌) ≠ (𝐹𝑦)) ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
4315, 42bitrd 278 . . . 4 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑥 ∈ {𝑋, 𝑌}∀𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦) ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
443, 43bitrid 282 . . 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, 45bitrid 282 1 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ (𝐹𝑋) ≠ (𝐹𝑌))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wne 2929  wral 3050  cdif 3941  {csn 4630  {cpr 4632  wf 6545  1-1wf1 6546  cfv 6549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fv 6557
This theorem is referenced by:  usgr2trlncl  29646
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