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Theorem iunopeqop 4951
Description: Implication of an ordered pair being equal to an indexed union of singletons of ordered pairs. (Contributed by AV, 20-Sep-2020.)
Hypotheses
Ref Expression
iunopeqop.b 𝐵 ∈ V
iunopeqop.c 𝐶 ∈ V
iunopeqop.d 𝐷 ∈ V
Assertion
Ref Expression
iunopeqop (𝐴 ≠ ∅ → ( 𝑥𝐴 {⟨𝑥, 𝐵⟩} = ⟨𝐶, 𝐷⟩ → ∃𝑧 𝐴 = {𝑧}))
Distinct variable groups:   𝑥,𝐴,𝑧   𝑥,𝐶   𝑥,𝐷
Allowed substitution hints:   𝐵(𝑥,𝑧)   𝐶(𝑧)   𝐷(𝑧)

Proof of Theorem iunopeqop
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 n0snor2el 4339 . 2 (𝐴 ≠ ∅ → (∃𝑥𝐴𝑦𝐴 𝑥𝑦 ∨ ∃𝑧 𝐴 = {𝑧}))
2 nfiu1 4523 . . . . . 6 𝑥 𝑥𝐴 {⟨𝑥, 𝐵⟩}
32nfeq1 2774 . . . . 5 𝑥 𝑥𝐴 {⟨𝑥, 𝐵⟩} = ⟨𝐶, 𝐷
4 nfv 1840 . . . . 5 𝑥𝑧 𝐴 = {𝑧}
53, 4nfim 1822 . . . 4 𝑥( 𝑥𝐴 {⟨𝑥, 𝐵⟩} = ⟨𝐶, 𝐷⟩ → ∃𝑧 𝐴 = {𝑧})
6 ssiun2 4536 . . . . . . 7 (𝑥𝐴 → {⟨𝑥, 𝐵⟩} ⊆ 𝑥𝐴 {⟨𝑥, 𝐵⟩})
7 nfcv 2761 . . . . . . . 8 𝑥𝑦
8 nfcsb1v 3535 . . . . . . . . . . 11 𝑥𝑦 / 𝑥𝐵
97, 8nfop 4393 . . . . . . . . . 10 𝑥𝑦, 𝑦 / 𝑥𝐵
109nfsn 4220 . . . . . . . . 9 𝑥{⟨𝑦, 𝑦 / 𝑥𝐵⟩}
1110, 2nfss 3581 . . . . . . . 8 𝑥{⟨𝑦, 𝑦 / 𝑥𝐵⟩} ⊆ 𝑥𝐴 {⟨𝑥, 𝐵⟩}
12 id 22 . . . . . . . . . . 11 (𝑥 = 𝑦𝑥 = 𝑦)
13 csbeq1a 3528 . . . . . . . . . . 11 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
1412, 13opeq12d 4385 . . . . . . . . . 10 (𝑥 = 𝑦 → ⟨𝑥, 𝐵⟩ = ⟨𝑦, 𝑦 / 𝑥𝐵⟩)
1514sneqd 4167 . . . . . . . . 9 (𝑥 = 𝑦 → {⟨𝑥, 𝐵⟩} = {⟨𝑦, 𝑦 / 𝑥𝐵⟩})
1615sseq1d 3617 . . . . . . . 8 (𝑥 = 𝑦 → ({⟨𝑥, 𝐵⟩} ⊆ 𝑥𝐴 {⟨𝑥, 𝐵⟩} ↔ {⟨𝑦, 𝑦 / 𝑥𝐵⟩} ⊆ 𝑥𝐴 {⟨𝑥, 𝐵⟩}))
177, 11, 16, 6vtoclgaf 3261 . . . . . . 7 (𝑦𝐴 → {⟨𝑦, 𝑦 / 𝑥𝐵⟩} ⊆ 𝑥𝐴 {⟨𝑥, 𝐵⟩})
186, 17anim12i 589 . . . . . 6 ((𝑥𝐴𝑦𝐴) → ({⟨𝑥, 𝐵⟩} ⊆ 𝑥𝐴 {⟨𝑥, 𝐵⟩} ∧ {⟨𝑦, 𝑦 / 𝑥𝐵⟩} ⊆ 𝑥𝐴 {⟨𝑥, 𝐵⟩}))
19 unss 3771 . . . . . . 7 (({⟨𝑥, 𝐵⟩} ⊆ 𝑥𝐴 {⟨𝑥, 𝐵⟩} ∧ {⟨𝑦, 𝑦 / 𝑥𝐵⟩} ⊆ 𝑥𝐴 {⟨𝑥, 𝐵⟩}) ↔ ({⟨𝑥, 𝐵⟩} ∪ {⟨𝑦, 𝑦 / 𝑥𝐵⟩}) ⊆ 𝑥𝐴 {⟨𝑥, 𝐵⟩})
20 sseq2 3612 . . . . . . . . 9 ( 𝑥𝐴 {⟨𝑥, 𝐵⟩} = ⟨𝐶, 𝐷⟩ → (({⟨𝑥, 𝐵⟩} ∪ {⟨𝑦, 𝑦 / 𝑥𝐵⟩}) ⊆ 𝑥𝐴 {⟨𝑥, 𝐵⟩} ↔ ({⟨𝑥, 𝐵⟩} ∪ {⟨𝑦, 𝑦 / 𝑥𝐵⟩}) ⊆ ⟨𝐶, 𝐷⟩))
21 df-pr 4158 . . . . . . . . . . . 12 {⟨𝑥, 𝐵⟩, ⟨𝑦, 𝑦 / 𝑥𝐵⟩} = ({⟨𝑥, 𝐵⟩} ∪ {⟨𝑦, 𝑦 / 𝑥𝐵⟩})
2221eqcomi 2630 . . . . . . . . . . 11 ({⟨𝑥, 𝐵⟩} ∪ {⟨𝑦, 𝑦 / 𝑥𝐵⟩}) = {⟨𝑥, 𝐵⟩, ⟨𝑦, 𝑦 / 𝑥𝐵⟩}
2322sseq1i 3614 . . . . . . . . . 10 (({⟨𝑥, 𝐵⟩} ∪ {⟨𝑦, 𝑦 / 𝑥𝐵⟩}) ⊆ ⟨𝐶, 𝐷⟩ ↔ {⟨𝑥, 𝐵⟩, ⟨𝑦, 𝑦 / 𝑥𝐵⟩} ⊆ ⟨𝐶, 𝐷⟩)
24 vex 3193 . . . . . . . . . . . 12 𝑥 ∈ V
25 iunopeqop.b . . . . . . . . . . . 12 𝐵 ∈ V
26 vex 3193 . . . . . . . . . . . 12 𝑦 ∈ V
2725csbex 4763 . . . . . . . . . . . 12 𝑦 / 𝑥𝐵 ∈ V
28 iunopeqop.c . . . . . . . . . . . 12 𝐶 ∈ V
29 iunopeqop.d . . . . . . . . . . . 12 𝐷 ∈ V
3024, 25, 26, 27, 28, 29propssopi 4941 . . . . . . . . . . 11 ({⟨𝑥, 𝐵⟩, ⟨𝑦, 𝑦 / 𝑥𝐵⟩} ⊆ ⟨𝐶, 𝐷⟩ → 𝑥 = 𝑦)
31 eqneqall 2801 . . . . . . . . . . 11 (𝑥 = 𝑦 → (𝑥𝑦 → ((𝑥𝐴𝑦𝐴) → ∃𝑧 𝐴 = {𝑧})))
3230, 31syl 17 . . . . . . . . . 10 ({⟨𝑥, 𝐵⟩, ⟨𝑦, 𝑦 / 𝑥𝐵⟩} ⊆ ⟨𝐶, 𝐷⟩ → (𝑥𝑦 → ((𝑥𝐴𝑦𝐴) → ∃𝑧 𝐴 = {𝑧})))
3323, 32sylbi 207 . . . . . . . . 9 (({⟨𝑥, 𝐵⟩} ∪ {⟨𝑦, 𝑦 / 𝑥𝐵⟩}) ⊆ ⟨𝐶, 𝐷⟩ → (𝑥𝑦 → ((𝑥𝐴𝑦𝐴) → ∃𝑧 𝐴 = {𝑧})))
3420, 33syl6bi 243 . . . . . . . 8 ( 𝑥𝐴 {⟨𝑥, 𝐵⟩} = ⟨𝐶, 𝐷⟩ → (({⟨𝑥, 𝐵⟩} ∪ {⟨𝑦, 𝑦 / 𝑥𝐵⟩}) ⊆ 𝑥𝐴 {⟨𝑥, 𝐵⟩} → (𝑥𝑦 → ((𝑥𝐴𝑦𝐴) → ∃𝑧 𝐴 = {𝑧}))))
3534com14 96 . . . . . . 7 ((𝑥𝐴𝑦𝐴) → (({⟨𝑥, 𝐵⟩} ∪ {⟨𝑦, 𝑦 / 𝑥𝐵⟩}) ⊆ 𝑥𝐴 {⟨𝑥, 𝐵⟩} → (𝑥𝑦 → ( 𝑥𝐴 {⟨𝑥, 𝐵⟩} = ⟨𝐶, 𝐷⟩ → ∃𝑧 𝐴 = {𝑧}))))
3619, 35syl5bi 232 . . . . . 6 ((𝑥𝐴𝑦𝐴) → (({⟨𝑥, 𝐵⟩} ⊆ 𝑥𝐴 {⟨𝑥, 𝐵⟩} ∧ {⟨𝑦, 𝑦 / 𝑥𝐵⟩} ⊆ 𝑥𝐴 {⟨𝑥, 𝐵⟩}) → (𝑥𝑦 → ( 𝑥𝐴 {⟨𝑥, 𝐵⟩} = ⟨𝐶, 𝐷⟩ → ∃𝑧 𝐴 = {𝑧}))))
3718, 36mpd 15 . . . . 5 ((𝑥𝐴𝑦𝐴) → (𝑥𝑦 → ( 𝑥𝐴 {⟨𝑥, 𝐵⟩} = ⟨𝐶, 𝐷⟩ → ∃𝑧 𝐴 = {𝑧})))
3837rexlimdva 3026 . . . 4 (𝑥𝐴 → (∃𝑦𝐴 𝑥𝑦 → ( 𝑥𝐴 {⟨𝑥, 𝐵⟩} = ⟨𝐶, 𝐷⟩ → ∃𝑧 𝐴 = {𝑧})))
395, 38rexlimi 3019 . . 3 (∃𝑥𝐴𝑦𝐴 𝑥𝑦 → ( 𝑥𝐴 {⟨𝑥, 𝐵⟩} = ⟨𝐶, 𝐷⟩ → ∃𝑧 𝐴 = {𝑧}))
40 ax-1 6 . . 3 (∃𝑧 𝐴 = {𝑧} → ( 𝑥𝐴 {⟨𝑥, 𝐵⟩} = ⟨𝐶, 𝐷⟩ → ∃𝑧 𝐴 = {𝑧}))
4139, 40jaoi 394 . 2 ((∃𝑥𝐴𝑦𝐴 𝑥𝑦 ∨ ∃𝑧 𝐴 = {𝑧}) → ( 𝑥𝐴 {⟨𝑥, 𝐵⟩} = ⟨𝐶, 𝐷⟩ → ∃𝑧 𝐴 = {𝑧}))
421, 41syl 17 1 (𝐴 ≠ ∅ → ( 𝑥𝐴 {⟨𝑥, 𝐵⟩} = ⟨𝐶, 𝐷⟩ → ∃𝑧 𝐴 = {𝑧}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 383  wa 384   = wceq 1480  wex 1701  wcel 1987  wne 2790  wrex 2909  Vcvv 3190  csb 3519  cun 3558  wss 3560  c0 3897  {csn 4155  {cpr 4157  cop 4161   ciun 4492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-iun 4494
This theorem is referenced by:  funopsn  6378
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