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Theorem ixpsnf1o 6630
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 snexg 4108 . . . 4 (𝐼𝑉 → {𝐼} ∈ V)
3 vex 2689 . . . . 5 𝑥 ∈ V
43snex 4109 . . . 4 {𝑥} ∈ V
5 xpexg 4653 . . . 4 (({𝐼} ∈ V ∧ {𝑥} ∈ V) → ({𝐼} × {𝑥}) ∈ V)
62, 4, 5sylancl 409 . . 3 (𝐼𝑉 → ({𝐼} × {𝑥}) ∈ V)
76adantr 274 . 2 ((𝐼𝑉𝑥𝐴) → ({𝐼} × {𝑥}) ∈ V)
8 vex 2689 . . . . 5 𝑎 ∈ V
98rnex 4806 . . . 4 ran 𝑎 ∈ V
109uniex 4359 . . 3 ran 𝑎 ∈ V
1110a1i 9 . 2 ((𝐼𝑉𝑎X𝑦 ∈ {𝐼}𝐴) → ran 𝑎 ∈ V)
12 sneq 3538 . . . . . 6 (𝑏 = 𝐼 → {𝑏} = {𝐼})
1312xpeq1d 4562 . . . . 5 (𝑏 = 𝐼 → ({𝑏} × {𝑥}) = ({𝐼} × {𝑥}))
1413eqeq2d 2151 . . . 4 (𝑏 = 𝐼 → (𝑎 = ({𝑏} × {𝑥}) ↔ 𝑎 = ({𝐼} × {𝑥})))
1514anbi2d 459 . . 3 (𝑏 = 𝐼 → ((𝑥𝐴𝑎 = ({𝑏} × {𝑥})) ↔ (𝑥𝐴𝑎 = ({𝐼} × {𝑥}))))
16 elixpsn 6629 . . . . . 6 (𝑏 ∈ V → (𝑎X𝑦 ∈ {𝑏}𝐴 ↔ ∃𝑐𝐴 𝑎 = {⟨𝑏, 𝑐⟩}))
1716elv 2690 . . . . 5 (𝑎X𝑦 ∈ {𝑏}𝐴 ↔ ∃𝑐𝐴 𝑎 = {⟨𝑏, 𝑐⟩})
1812ixpeq1d 6604 . . . . . 6 (𝑏 = 𝐼X𝑦 ∈ {𝑏}𝐴 = X𝑦 ∈ {𝐼}𝐴)
1918eleq2d 2209 . . . . 5 (𝑏 = 𝐼 → (𝑎X𝑦 ∈ {𝑏}𝐴𝑎X𝑦 ∈ {𝐼}𝐴))
2017, 19bitr3id 193 . . . 4 (𝑏 = 𝐼 → (∃𝑐𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ↔ 𝑎X𝑦 ∈ {𝐼}𝐴))
2120anbi1d 460 . . 3 (𝑏 = 𝐼 → ((∃𝑐𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ran 𝑎) ↔ (𝑎X𝑦 ∈ {𝐼}𝐴𝑥 = ran 𝑎)))
22 vex 2689 . . . . . . 7 𝑏 ∈ V
2322, 3xpsn 5596 . . . . . 6 ({𝑏} × {𝑥}) = {⟨𝑏, 𝑥⟩}
2423eqeq2i 2150 . . . . 5 (𝑎 = ({𝑏} × {𝑥}) ↔ 𝑎 = {⟨𝑏, 𝑥⟩})
2524anbi2i 452 . . . 4 ((𝑥𝐴𝑎 = ({𝑏} × {𝑥})) ↔ (𝑥𝐴𝑎 = {⟨𝑏, 𝑥⟩}))
26 eqid 2139 . . . . . . . . 9 {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑥⟩}
27 opeq2 3706 . . . . . . . . . . 11 (𝑐 = 𝑥 → ⟨𝑏, 𝑐⟩ = ⟨𝑏, 𝑥⟩)
2827sneqd 3540 . . . . . . . . . 10 (𝑐 = 𝑥 → {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩})
2928rspceeqv 2807 . . . . . . . . 9 ((𝑥𝐴 ∧ {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑥⟩}) → ∃𝑐𝐴 {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩})
3026, 29mpan2 421 . . . . . . . 8 (𝑥𝐴 → ∃𝑐𝐴 {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩})
3122, 3op2nda 5023 . . . . . . . . 9 ran {⟨𝑏, 𝑥⟩} = 𝑥
3231eqcomi 2143 . . . . . . . 8 𝑥 = ran {⟨𝑏, 𝑥⟩}
3330, 32jctir 311 . . . . . . 7 (𝑥𝐴 → (∃𝑐𝐴 {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ran {⟨𝑏, 𝑥⟩}))
34 eqeq1 2146 . . . . . . . . 9 (𝑎 = {⟨𝑏, 𝑥⟩} → (𝑎 = {⟨𝑏, 𝑐⟩} ↔ {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩}))
3534rexbidv 2438 . . . . . . . 8 (𝑎 = {⟨𝑏, 𝑥⟩} → (∃𝑐𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ↔ ∃𝑐𝐴 {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩}))
36 rneq 4766 . . . . . . . . . 10 (𝑎 = {⟨𝑏, 𝑥⟩} → ran 𝑎 = ran {⟨𝑏, 𝑥⟩})
3736unieqd 3747 . . . . . . . . 9 (𝑎 = {⟨𝑏, 𝑥⟩} → ran 𝑎 = ran {⟨𝑏, 𝑥⟩})
3837eqeq2d 2151 . . . . . . . 8 (𝑎 = {⟨𝑏, 𝑥⟩} → (𝑥 = ran 𝑎𝑥 = ran {⟨𝑏, 𝑥⟩}))
3935, 38anbi12d 464 . . . . . . 7 (𝑎 = {⟨𝑏, 𝑥⟩} → ((∃𝑐𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ran 𝑎) ↔ (∃𝑐𝐴 {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ran {⟨𝑏, 𝑥⟩})))
4033, 39syl5ibrcom 156 . . . . . 6 (𝑥𝐴 → (𝑎 = {⟨𝑏, 𝑥⟩} → (∃𝑐𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ran 𝑎)))
4140imp 123 . . . . 5 ((𝑥𝐴𝑎 = {⟨𝑏, 𝑥⟩}) → (∃𝑐𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ran 𝑎))
42 vex 2689 . . . . . . . . . . 11 𝑐 ∈ V
4322, 42op2nda 5023 . . . . . . . . . 10 ran {⟨𝑏, 𝑐⟩} = 𝑐
4443eqeq2i 2150 . . . . . . . . 9 (𝑥 = ran {⟨𝑏, 𝑐⟩} ↔ 𝑥 = 𝑐)
45 eqidd 2140 . . . . . . . . . . 11 (𝑐𝐴 → {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑐⟩})
4645ancli 321 . . . . . . . . . 10 (𝑐𝐴 → (𝑐𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑐⟩}))
47 eleq1w 2200 . . . . . . . . . . 11 (𝑥 = 𝑐 → (𝑥𝐴𝑐𝐴))
48 opeq2 3706 . . . . . . . . . . . . 13 (𝑥 = 𝑐 → ⟨𝑏, 𝑥⟩ = ⟨𝑏, 𝑐⟩)
4948sneqd 3540 . . . . . . . . . . . 12 (𝑥 = 𝑐 → {⟨𝑏, 𝑥⟩} = {⟨𝑏, 𝑐⟩})
5049eqeq2d 2151 . . . . . . . . . . 11 (𝑥 = 𝑐 → ({⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩} ↔ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑐⟩}))
5147, 50anbi12d 464 . . . . . . . . . 10 (𝑥 = 𝑐 → ((𝑥𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩}) ↔ (𝑐𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑐⟩})))
5246, 51syl5ibrcom 156 . . . . . . . . 9 (𝑐𝐴 → (𝑥 = 𝑐 → (𝑥𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩})))
5344, 52syl5bi 151 . . . . . . . 8 (𝑐𝐴 → (𝑥 = ran {⟨𝑏, 𝑐⟩} → (𝑥𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩})))
54 rneq 4766 . . . . . . . . . . 11 (𝑎 = {⟨𝑏, 𝑐⟩} → ran 𝑎 = ran {⟨𝑏, 𝑐⟩})
5554unieqd 3747 . . . . . . . . . 10 (𝑎 = {⟨𝑏, 𝑐⟩} → ran 𝑎 = ran {⟨𝑏, 𝑐⟩})
5655eqeq2d 2151 . . . . . . . . 9 (𝑎 = {⟨𝑏, 𝑐⟩} → (𝑥 = ran 𝑎𝑥 = ran {⟨𝑏, 𝑐⟩}))
57 eqeq1 2146 . . . . . . . . . 10 (𝑎 = {⟨𝑏, 𝑐⟩} → (𝑎 = {⟨𝑏, 𝑥⟩} ↔ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩}))
5857anbi2d 459 . . . . . . . . 9 (𝑎 = {⟨𝑏, 𝑐⟩} → ((𝑥𝐴𝑎 = {⟨𝑏, 𝑥⟩}) ↔ (𝑥𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩})))
5956, 58imbi12d 233 . . . . . . . 8 (𝑎 = {⟨𝑏, 𝑐⟩} → ((𝑥 = ran 𝑎 → (𝑥𝐴𝑎 = {⟨𝑏, 𝑥⟩})) ↔ (𝑥 = ran {⟨𝑏, 𝑐⟩} → (𝑥𝐴 ∧ {⟨𝑏, 𝑐⟩} = {⟨𝑏, 𝑥⟩}))))
6053, 59syl5ibrcom 156 . . . . . . 7 (𝑐𝐴 → (𝑎 = {⟨𝑏, 𝑐⟩} → (𝑥 = ran 𝑎 → (𝑥𝐴𝑎 = {⟨𝑏, 𝑥⟩}))))
6160rexlimiv 2543 . . . . . 6 (∃𝑐𝐴 𝑎 = {⟨𝑏, 𝑐⟩} → (𝑥 = ran 𝑎 → (𝑥𝐴𝑎 = {⟨𝑏, 𝑥⟩})))
6261imp 123 . . . . 5 ((∃𝑐𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ran 𝑎) → (𝑥𝐴𝑎 = {⟨𝑏, 𝑥⟩}))
6341, 62impbii 125 . . . 4 ((𝑥𝐴𝑎 = {⟨𝑏, 𝑥⟩}) ↔ (∃𝑐𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ran 𝑎))
6425, 63bitri 183 . . 3 ((𝑥𝐴𝑎 = ({𝑏} × {𝑥})) ↔ (∃𝑐𝐴 𝑎 = {⟨𝑏, 𝑐⟩} ∧ 𝑥 = ran 𝑎))
6515, 21, 64vtoclbg 2747 . 2 (𝐼𝑉 → ((𝑥𝐴𝑎 = ({𝐼} × {𝑥})) ↔ (𝑎X𝑦 ∈ {𝐼}𝐴𝑥 = ran 𝑎)))
661, 7, 11, 65f1od 5973 1 (𝐼𝑉𝐹:𝐴1-1-ontoX𝑦 ∈ {𝐼}𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1331  wcel 1480  wrex 2417  Vcvv 2686  {csn 3527  cop 3530   cuni 3736  cmpt 3989   × cxp 4537  ran crn 4540  1-1-ontowf1o 5122  Xcixp 6592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-reu 2423  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ixp 6593
This theorem is referenced by:  mapsnf1o  6631
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