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Theorem f12dfv 6721
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 6720 . 2 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦)))
2 f12dfv.a . . . . 5 𝐴 = {𝑋, 𝑌}
32raleqi 3290 . . . 4 (∀𝑥𝐴𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦) ↔ ∀𝑥 ∈ {𝑋, 𝑌}∀𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦))
4 sneq 4344 . . . . . . . . 9 (𝑥 = 𝑋 → {𝑥} = {𝑋})
54difeq2d 3890 . . . . . . . 8 (𝑥 = 𝑋 → (𝐴 ∖ {𝑥}) = (𝐴 ∖ {𝑋}))
6 fveq2 6375 . . . . . . . . 9 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
76neeq1d 2996 . . . . . . . 8 (𝑥 = 𝑋 → ((𝐹𝑥) ≠ (𝐹𝑦) ↔ (𝐹𝑋) ≠ (𝐹𝑦)))
85, 7raleqbidv 3300 . . . . . . 7 (𝑥 = 𝑋 → (∀𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦) ↔ ∀𝑦 ∈ (𝐴 ∖ {𝑋})(𝐹𝑋) ≠ (𝐹𝑦)))
9 sneq 4344 . . . . . . . . 9 (𝑥 = 𝑌 → {𝑥} = {𝑌})
109difeq2d 3890 . . . . . . . 8 (𝑥 = 𝑌 → (𝐴 ∖ {𝑥}) = (𝐴 ∖ {𝑌}))
11 fveq2 6375 . . . . . . . . 9 (𝑥 = 𝑌 → (𝐹𝑥) = (𝐹𝑌))
1211neeq1d 2996 . . . . . . . 8 (𝑥 = 𝑌 → ((𝐹𝑥) ≠ (𝐹𝑦) ↔ (𝐹𝑌) ≠ (𝐹𝑦)))
1310, 12raleqbidv 3300 . . . . . . 7 (𝑥 = 𝑌 → (∀𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦) ↔ ∀𝑦 ∈ (𝐴 ∖ {𝑌})(𝐹𝑌) ≠ (𝐹𝑦)))
148, 13ralprg 4390 . . . . . 6 ((𝑋𝑈𝑌𝑉) → (∀𝑥 ∈ {𝑋, 𝑌}∀𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦) ↔ (∀𝑦 ∈ (𝐴 ∖ {𝑋})(𝐹𝑋) ≠ (𝐹𝑦) ∧ ∀𝑦 ∈ (𝐴 ∖ {𝑌})(𝐹𝑌) ≠ (𝐹𝑦))))
1514adantr 472 . . . . 5 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑥 ∈ {𝑋, 𝑌}∀𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦) ↔ (∀𝑦 ∈ (𝐴 ∖ {𝑋})(𝐹𝑋) ≠ (𝐹𝑦) ∧ ∀𝑦 ∈ (𝐴 ∖ {𝑌})(𝐹𝑌) ≠ (𝐹𝑦))))
162difeq1i 3886 . . . . . . . . . . 11 (𝐴 ∖ {𝑋}) = ({𝑋, 𝑌} ∖ {𝑋})
17 difprsn1 4485 . . . . . . . . . . 11 (𝑋𝑌 → ({𝑋, 𝑌} ∖ {𝑋}) = {𝑌})
1816, 17syl5eq 2811 . . . . . . . . . 10 (𝑋𝑌 → (𝐴 ∖ {𝑋}) = {𝑌})
1918adantl 473 . . . . . . . . 9 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (𝐴 ∖ {𝑋}) = {𝑌})
2019raleqdv 3292 . . . . . . . 8 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑦 ∈ (𝐴 ∖ {𝑋})(𝐹𝑋) ≠ (𝐹𝑦) ↔ ∀𝑦 ∈ {𝑌} (𝐹𝑋) ≠ (𝐹𝑦)))
21 fveq2 6375 . . . . . . . . . . . 12 (𝑦 = 𝑌 → (𝐹𝑦) = (𝐹𝑌))
2221neeq2d 2997 . . . . . . . . . . 11 (𝑦 = 𝑌 → ((𝐹𝑋) ≠ (𝐹𝑦) ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
2322ralsng 4375 . . . . . . . . . 10 (𝑌𝑉 → (∀𝑦 ∈ {𝑌} (𝐹𝑋) ≠ (𝐹𝑦) ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
2423adantl 473 . . . . . . . . 9 ((𝑋𝑈𝑌𝑉) → (∀𝑦 ∈ {𝑌} (𝐹𝑋) ≠ (𝐹𝑦) ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
2524adantr 472 . . . . . . . 8 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑦 ∈ {𝑌} (𝐹𝑋) ≠ (𝐹𝑦) ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
2620, 25bitrd 270 . . . . . . 7 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑦 ∈ (𝐴 ∖ {𝑋})(𝐹𝑋) ≠ (𝐹𝑦) ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
272difeq1i 3886 . . . . . . . . . . 11 (𝐴 ∖ {𝑌}) = ({𝑋, 𝑌} ∖ {𝑌})
28 difprsn2 4486 . . . . . . . . . . 11 (𝑋𝑌 → ({𝑋, 𝑌} ∖ {𝑌}) = {𝑋})
2927, 28syl5eq 2811 . . . . . . . . . 10 (𝑋𝑌 → (𝐴 ∖ {𝑌}) = {𝑋})
3029adantl 473 . . . . . . . . 9 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (𝐴 ∖ {𝑌}) = {𝑋})
3130raleqdv 3292 . . . . . . . 8 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑦 ∈ (𝐴 ∖ {𝑌})(𝐹𝑌) ≠ (𝐹𝑦) ↔ ∀𝑦 ∈ {𝑋} (𝐹𝑌) ≠ (𝐹𝑦)))
32 fveq2 6375 . . . . . . . . . . . 12 (𝑦 = 𝑋 → (𝐹𝑦) = (𝐹𝑋))
3332neeq2d 2997 . . . . . . . . . . 11 (𝑦 = 𝑋 → ((𝐹𝑌) ≠ (𝐹𝑦) ↔ (𝐹𝑌) ≠ (𝐹𝑋)))
3433ralsng 4375 . . . . . . . . . 10 (𝑋𝑈 → (∀𝑦 ∈ {𝑋} (𝐹𝑌) ≠ (𝐹𝑦) ↔ (𝐹𝑌) ≠ (𝐹𝑋)))
3534adantr 472 . . . . . . . . 9 ((𝑋𝑈𝑌𝑉) → (∀𝑦 ∈ {𝑋} (𝐹𝑌) ≠ (𝐹𝑦) ↔ (𝐹𝑌) ≠ (𝐹𝑋)))
3635adantr 472 . . . . . . . 8 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑦 ∈ {𝑋} (𝐹𝑌) ≠ (𝐹𝑦) ↔ (𝐹𝑌) ≠ (𝐹𝑋)))
3731, 36bitrd 270 . . . . . . 7 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑦 ∈ (𝐴 ∖ {𝑌})(𝐹𝑌) ≠ (𝐹𝑦) ↔ (𝐹𝑌) ≠ (𝐹𝑋)))
3826, 37anbi12d 624 . . . . . 6 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → ((∀𝑦 ∈ (𝐴 ∖ {𝑋})(𝐹𝑋) ≠ (𝐹𝑦) ∧ ∀𝑦 ∈ (𝐴 ∖ {𝑌})(𝐹𝑌) ≠ (𝐹𝑦)) ↔ ((𝐹𝑋) ≠ (𝐹𝑌) ∧ (𝐹𝑌) ≠ (𝐹𝑋))))
39 necom 2990 . . . . . . . 8 ((𝐹𝑋) ≠ (𝐹𝑌) ↔ (𝐹𝑌) ≠ (𝐹𝑋))
4039biimpi 207 . . . . . . 7 ((𝐹𝑋) ≠ (𝐹𝑌) → (𝐹𝑌) ≠ (𝐹𝑋))
4140pm4.71i 555 . . . . . 6 ((𝐹𝑋) ≠ (𝐹𝑌) ↔ ((𝐹𝑋) ≠ (𝐹𝑌) ∧ (𝐹𝑌) ≠ (𝐹𝑋)))
4238, 41syl6bbr 280 . . . . 5 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → ((∀𝑦 ∈ (𝐴 ∖ {𝑋})(𝐹𝑋) ≠ (𝐹𝑦) ∧ ∀𝑦 ∈ (𝐴 ∖ {𝑌})(𝐹𝑌) ≠ (𝐹𝑦)) ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
4315, 42bitrd 270 . . . 4 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑥 ∈ {𝑋, 𝑌}∀𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦) ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
443, 43syl5bb 274 . . 3 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (∀𝑥𝐴𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦) ↔ (𝐹𝑋) ≠ (𝐹𝑌)))
4544anbi2d 622 . 2 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → ((𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦)) ↔ (𝐹:𝐴𝐵 ∧ (𝐹𝑋) ≠ (𝐹𝑌))))
461, 45syl5bb 274 1 (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ (𝐹𝑋) ≠ (𝐹𝑌))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wne 2937  wral 3055  cdif 3729  {csn 4334  {cpr 4336  wf 6064  1-1wf1 6065  cfv 6068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fv 6076
This theorem is referenced by:  usgr2trlncl  26948
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