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Theorem ixpsnf1o 7933
Description: A bijection between a class and single-point functions to it. (Contributed by Stefan O'Rear, 24-Jan-2015.)
Hypothesis
Ref Expression
ixpsnf1o.f 𝐹 = (𝑥𝐴 ↦ ({𝐼} × {𝑥}))
Assertion
Ref Expression
ixpsnf1o (𝐼𝑉𝐹:𝐴1-1-ontoX𝑦 ∈ {𝐼}𝐴)
Distinct variable groups:   𝑥,𝐼,𝑦   𝑥,𝐴,𝑦   𝑥,𝑉,𝑦   𝑦,𝐹
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem ixpsnf1o
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ixpsnf1o.f . 2 𝐹 = (𝑥𝐴 ↦ ({𝐼} × {𝑥}))
2 snex 4899 . . . 4 {𝐼} ∈ V
3 snex 4899 . . . 4 {𝑥} ∈ V
42, 3xpex 6947 . . 3 ({𝐼} × {𝑥}) ∈ V
54a1i 11 . 2 ((𝐼𝑉𝑥𝐴) → ({𝐼} × {𝑥}) ∈ V)
6 vex 3198 . . . . 5 𝑎 ∈ V
76rnex 7085 . . . 4 ran 𝑎 ∈ V
87uniex 6938 . . 3 ran 𝑎 ∈ V
98a1i 11 . 2 ((𝐼𝑉𝑎X𝑦 ∈ {𝐼}𝐴) → ran 𝑎 ∈ V)
10 sneq 4178 . . . . . 6 (𝑏 = 𝐼 → {𝑏} = {𝐼})
1110xpeq1d 5128 . . . . 5 (𝑏 = 𝐼 → ({𝑏} × {𝑥}) = ({𝐼} × {𝑥}))
1211eqeq2d 2630 . . . 4 (𝑏 = 𝐼 → (𝑎 = ({𝑏} × {𝑥}) ↔ 𝑎 = ({𝐼} × {𝑥})))
1312anbi2d 739 . . 3 (𝑏 = 𝐼 → ((𝑥𝐴𝑎 = ({𝑏} × {𝑥})) ↔ (𝑥𝐴𝑎 = ({𝐼} × {𝑥}))))
14 vex 3198 . . . . . 6 𝑏 ∈ V
15 elixpsn 7932 . . . . . 6 (𝑏 ∈ V → (𝑎X𝑦 ∈ {𝑏}𝐴 ↔ ∃𝑐𝐴 𝑎 = {⟨𝑏, 𝑐⟩}))
1614, 15ax-mp 5 . . . . 5 (𝑎X𝑦 ∈ {𝑏}𝐴 ↔ ∃𝑐𝐴 𝑎 = {⟨𝑏, 𝑐⟩})
1710ixpeq1d 7905 . . . . . 6 (𝑏 = 𝐼X𝑦 ∈ {𝑏}𝐴 = X𝑦 ∈ {𝐼}𝐴)
1817eleq2d 2685 . . . . 5 (𝑏 = 𝐼 → (𝑎X𝑦 ∈ {𝑏}𝐴𝑎X𝑦 ∈ {𝐼}𝐴))
1916, 18syl5bbr 274 . . . 4 (𝑏 = 𝐼 → (∃𝑐𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ↔ 𝑎X𝑦 ∈ {𝐼}𝐴))
2019anbi1d 740 . . 3 (𝑏 = 𝐼 → ((∃𝑐𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ran 𝑎) ↔ (𝑎X𝑦 ∈ {𝐼}𝐴𝑥 = ran 𝑎)))
21 vex 3198 . . . . . . 7 𝑥 ∈ V
2214, 21xpsn 6392 . . . . . 6 ({𝑏} × {𝑥}) = {⟨𝑏, 𝑥⟩}
2322eqeq2i 2632 . . . . 5 (𝑎 = ({𝑏} × {𝑥}) ↔ 𝑎 = {⟨𝑏, 𝑥⟩})
2423anbi2i 729 . . . 4 ((𝑥𝐴𝑎 = ({𝑏} × {𝑥})) ↔ (𝑥𝐴𝑎 = {⟨𝑏, 𝑥⟩}))
25 eqid 2620 . . . . . . . . 9 {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑥⟩}
26 opeq2 4394 . . . . . . . . . . . 12 (𝑐 = 𝑥 → ⟨𝑏, 𝑐⟩ = ⟨𝑏, 𝑥⟩)
2726sneqd 4180 . . . . . . . . . . 11 (𝑐 = 𝑥 → {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩})
2827eqeq2d 2630 . . . . . . . . . 10 (𝑐 = 𝑥 → ({⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩} ↔ {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑥⟩}))
2928rspcev 3304 . . . . . . . . 9 ((𝑥𝐴 ∧ {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑥⟩}) → ∃𝑐𝐴 {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩})
3025, 29mpan2 706 . . . . . . . 8 (𝑥𝐴 → ∃𝑐𝐴 {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩})
3114, 21op2nda 5608 . . . . . . . . 9 ran {⟨𝑏, 𝑥⟩} = 𝑥
3231eqcomi 2629 . . . . . . . 8 𝑥 = ran {⟨𝑏, 𝑥⟩}
3330, 32jctir 560 . . . . . . 7 (𝑥𝐴 → (∃𝑐𝐴 {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ran {⟨𝑏, 𝑥⟩}))
34 eqeq1 2624 . . . . . . . . 9 (𝑎 = {⟨𝑏, 𝑥⟩} → (𝑎 = {⟨𝑏, 𝑐⟩} ↔ {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩}))
3534rexbidv 3048 . . . . . . . 8 (𝑎 = {⟨𝑏, 𝑥⟩} → (∃𝑐𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ↔ ∃𝑐𝐴 {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩}))
36 rneq 5340 . . . . . . . . . 10 (𝑎 = {⟨𝑏, 𝑥⟩} → ran 𝑎 = ran {⟨𝑏, 𝑥⟩})
3736unieqd 4437 . . . . . . . . 9 (𝑎 = {⟨𝑏, 𝑥⟩} → ran 𝑎 = ran {⟨𝑏, 𝑥⟩})
3837eqeq2d 2630 . . . . . . . 8 (𝑎 = {⟨𝑏, 𝑥⟩} → (𝑥 = ran 𝑎𝑥 = ran {⟨𝑏, 𝑥⟩}))
3935, 38anbi12d 746 . . . . . . 7 (𝑎 = {⟨𝑏, 𝑥⟩} → ((∃𝑐𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ran 𝑎) ↔ (∃𝑐𝐴 {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ran {⟨𝑏, 𝑥⟩})))
4033, 39syl5ibrcom 237 . . . . . 6 (𝑥𝐴 → (𝑎 = {⟨𝑏, 𝑥⟩} → (∃𝑐𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ran 𝑎)))
4140imp 445 . . . . 5 ((𝑥𝐴𝑎 = {⟨𝑏, 𝑥⟩}) → (∃𝑐𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ran 𝑎))
42 vex 3198 . . . . . . . . . . 11 𝑐 ∈ V
4314, 42op2nda 5608 . . . . . . . . . 10 ran {⟨𝑏, 𝑐⟩} = 𝑐
4443eqeq2i 2632 . . . . . . . . 9 (𝑥 = ran {⟨𝑏, 𝑐⟩} ↔ 𝑥 = 𝑐)
45 eqidd 2621 . . . . . . . . . . 11 (𝑐𝐴 → {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑐⟩})
4645ancli 573 . . . . . . . . . 10 (𝑐𝐴 → (𝑐𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑐⟩}))
47 eleq1 2687 . . . . . . . . . . 11 (𝑥 = 𝑐 → (𝑥𝐴𝑐𝐴))
48 opeq2 4394 . . . . . . . . . . . . 13 (𝑥 = 𝑐 → ⟨𝑏, 𝑥⟩ = ⟨𝑏, 𝑐⟩)
4948sneqd 4180 . . . . . . . . . . . 12 (𝑥 = 𝑐 → {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩})
5049eqeq2d 2630 . . . . . . . . . . 11 (𝑥 = 𝑐 → ({⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩} ↔ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑐⟩}))
5147, 50anbi12d 746 . . . . . . . . . 10 (𝑥 = 𝑐 → ((𝑥𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩}) ↔ (𝑐𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑐⟩})))
5246, 51syl5ibrcom 237 . . . . . . . . 9 (𝑐𝐴 → (𝑥 = 𝑐 → (𝑥𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩})))
5344, 52syl5bi 232 . . . . . . . 8 (𝑐𝐴 → (𝑥 = ran {⟨𝑏, 𝑐⟩} → (𝑥𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩})))
54 rneq 5340 . . . . . . . . . . 11 (𝑎 = {⟨𝑏, 𝑐⟩} → ran 𝑎 = ran {⟨𝑏, 𝑐⟩})
5554unieqd 4437 . . . . . . . . . 10 (𝑎 = {⟨𝑏, 𝑐⟩} → ran 𝑎 = ran {⟨𝑏, 𝑐⟩})
5655eqeq2d 2630 . . . . . . . . 9 (𝑎 = {⟨𝑏, 𝑐⟩} → (𝑥 = ran 𝑎𝑥 = ran {⟨𝑏, 𝑐⟩}))
57 eqeq1 2624 . . . . . . . . . 10 (𝑎 = {⟨𝑏, 𝑐⟩} → (𝑎 = {⟨𝑏, 𝑥⟩} ↔ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩}))
5857anbi2d 739 . . . . . . . . 9 (𝑎 = {⟨𝑏, 𝑐⟩} → ((𝑥𝐴𝑎 = {⟨𝑏, 𝑥⟩}) ↔ (𝑥𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩})))
5956, 58imbi12d 334 . . . . . . . 8 (𝑎 = {⟨𝑏, 𝑐⟩} → ((𝑥 = ran 𝑎 → (𝑥𝐴𝑎 = {⟨𝑏, 𝑥⟩})) ↔ (𝑥 = ran {⟨𝑏, 𝑐⟩} → (𝑥𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩}))))
6053, 59syl5ibrcom 237 . . . . . . 7 (𝑐𝐴 → (𝑎 = {⟨𝑏, 𝑐⟩} → (𝑥 = ran 𝑎 → (𝑥𝐴𝑎 = {⟨𝑏, 𝑥⟩}))))
6160rexlimiv 3023 . . . . . 6 (∃𝑐𝐴 𝑎 = {⟨𝑏, 𝑐⟩} → (𝑥 = ran 𝑎 → (𝑥𝐴𝑎 = {⟨𝑏, 𝑥⟩})))
6261imp 445 . . . . 5 ((∃𝑐𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ran 𝑎) → (𝑥𝐴𝑎 = {⟨𝑏, 𝑥⟩}))
6341, 62impbii 199 . . . 4 ((𝑥𝐴𝑎 = {⟨𝑏, 𝑥⟩}) ↔ (∃𝑐𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ran 𝑎))
6424, 63bitri 264 . . 3 ((𝑥𝐴𝑎 = ({𝑏} × {𝑥})) ↔ (∃𝑐𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ran 𝑎))
6513, 20, 64vtoclbg 3262 . 2 (𝐼𝑉 → ((𝑥𝐴𝑎 = ({𝐼} × {𝑥})) ↔ (𝑎X𝑦 ∈ {𝐼}𝐴𝑥 = ran 𝑎)))
661, 5, 9, 65f1od 6870 1 (𝐼𝑉𝐹:𝐴1-1-ontoX𝑦 ∈ {𝐼}𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  wrex 2910  Vcvv 3195  {csn 4168  cop 4174   cuni 4427  cmpt 4720   × cxp 5102  ran crn 5105  1-1-ontowf1o 5875  Xcixp 7893
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ixp 7894
This theorem is referenced by:  mapsnf1o  7934
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