Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  iunopeqop Structured version   Visualization version   GIF version

Theorem iunopeqop 40236
 Description: Equivalence for an ordered pair 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 40228 . 2 (𝐴 ≠ ∅ → (∃𝑥𝐴𝑦𝐴 𝑥𝑦 ∨ ∃𝑧 𝐴 = {𝑧}))
2 nfiu1 4384 . . . . . 6 𝑥 𝑥𝐴 {⟨𝑥, 𝐵⟩}
32nfeq1 2668 . . . . 5 𝑥 𝑥𝐴 {⟨𝑥, 𝐵⟩} = ⟨𝐶, 𝐷
4 nfv 1796 . . . . 5 𝑥𝑧 𝐴 = {𝑧}
53, 4nfim 2051 . . . 4 𝑥( 𝑥𝐴 {⟨𝑥, 𝐵⟩} = ⟨𝐶, 𝐷⟩ → ∃𝑧 𝐴 = {𝑧})
6 ssiun2 4397 . . . . . . 7 (𝑥𝐴 → {⟨𝑥, 𝐵⟩} ⊆ 𝑥𝐴 {⟨𝑥, 𝐵⟩})
7 nfcv 2655 . . . . . . . 8 𝑥𝑦
8 nfcsb1v 3419 . . . . . . . . . . 11 𝑥𝑦 / 𝑥𝐵
97, 8nfop 4254 . . . . . . . . . 10 𝑥𝑦, 𝑦 / 𝑥𝐵
109nfsn 4092 . . . . . . . . 9 𝑥{⟨𝑦, 𝑦 / 𝑥𝐵⟩}
1110, 2nfss 3465 . . . . . . . 8 𝑥{⟨𝑦, 𝑦 / 𝑥𝐵⟩} ⊆ 𝑥𝐴 {⟨𝑥, 𝐵⟩}
12 id 22 . . . . . . . . . . 11 (𝑥 = 𝑦𝑥 = 𝑦)
13 csbeq1a 3412 . . . . . . . . . . 11 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
1412, 13opeq12d 4246 . . . . . . . . . 10 (𝑥 = 𝑦 → ⟨𝑥, 𝐵⟩ = ⟨𝑦, 𝑦 / 𝑥𝐵⟩)
1514sneqd 4040 . . . . . . . . 9 (𝑥 = 𝑦 → {⟨𝑥, 𝐵⟩} = {⟨𝑦, 𝑦 / 𝑥𝐵⟩})
1615sseq1d 3499 . . . . . . . 8 (𝑥 = 𝑦 → ({⟨𝑥, 𝐵⟩} ⊆ 𝑥𝐴 {⟨𝑥, 𝐵⟩} ↔ {⟨𝑦, 𝑦 / 𝑥𝐵⟩} ⊆ 𝑥𝐴 {⟨𝑥, 𝐵⟩}))
177, 11, 16, 6vtoclgaf 3148 . . . . . . 7 (𝑦𝐴 → {⟨𝑦, 𝑦 / 𝑥𝐵⟩} ⊆ 𝑥𝐴 {⟨𝑥, 𝐵⟩})
186, 17anim12i 587 . . . . . 6 ((𝑥𝐴𝑦𝐴) → ({⟨𝑥, 𝐵⟩} ⊆ 𝑥𝐴 {⟨𝑥, 𝐵⟩} ∧ {⟨𝑦, 𝑦 / 𝑥𝐵⟩} ⊆ 𝑥𝐴 {⟨𝑥, 𝐵⟩}))
19 unss 3653 . . . . . . 7 (({⟨𝑥, 𝐵⟩} ⊆ 𝑥𝐴 {⟨𝑥, 𝐵⟩} ∧ {⟨𝑦, 𝑦 / 𝑥𝐵⟩} ⊆ 𝑥𝐴 {⟨𝑥, 𝐵⟩}) ↔ ({⟨𝑥, 𝐵⟩} ∪ {⟨𝑦, 𝑦 / 𝑥𝐵⟩}) ⊆ 𝑥𝐴 {⟨𝑥, 𝐵⟩})
20 sseq2 3494 . . . . . . . . 9 ( 𝑥𝐴 {⟨𝑥, 𝐵⟩} = ⟨𝐶, 𝐷⟩ → (({⟨𝑥, 𝐵⟩} ∪ {⟨𝑦, 𝑦 / 𝑥𝐵⟩}) ⊆ 𝑥𝐴 {⟨𝑥, 𝐵⟩} ↔ ({⟨𝑥, 𝐵⟩} ∪ {⟨𝑦, 𝑦 / 𝑥𝐵⟩}) ⊆ ⟨𝐶, 𝐷⟩))
21 df-pr 4031 . . . . . . . . . . . 12 {⟨𝑥, 𝐵⟩, ⟨𝑦, 𝑦 / 𝑥𝐵⟩} = ({⟨𝑥, 𝐵⟩} ∪ {⟨𝑦, 𝑦 / 𝑥𝐵⟩})
2221eqcomi 2523 . . . . . . . . . . 11 ({⟨𝑥, 𝐵⟩} ∪ {⟨𝑦, 𝑦 / 𝑥𝐵⟩}) = {⟨𝑥, 𝐵⟩, ⟨𝑦, 𝑦 / 𝑥𝐵⟩}
2322sseq1i 3496 . . . . . . . . . 10 (({⟨𝑥, 𝐵⟩} ∪ {⟨𝑦, 𝑦 / 𝑥𝐵⟩}) ⊆ ⟨𝐶, 𝐷⟩ ↔ {⟨𝑥, 𝐵⟩, ⟨𝑦, 𝑦 / 𝑥𝐵⟩} ⊆ ⟨𝐶, 𝐷⟩)
24 vex 3080 . . . . . . . . . . . 12 𝑥 ∈ V
25 iunopeqop.b . . . . . . . . . . . 12 𝐵 ∈ V
26 vex 3080 . . . . . . . . . . . 12 𝑦 ∈ V
2725csbex 4620 . . . . . . . . . . . 12 𝑦 / 𝑥𝐵 ∈ V
28 iunopeqop.c . . . . . . . . . . . 12 𝐶 ∈ V
29 iunopeqop.d . . . . . . . . . . . 12 𝐷 ∈ V
3024, 25, 26, 27, 28, 29propssopi 40232 . . . . . . . . . . 11 ({⟨𝑥, 𝐵⟩, ⟨𝑦, 𝑦 / 𝑥𝐵⟩} ⊆ ⟨𝐶, 𝐷⟩ → 𝑥 = 𝑦)
31 eqneqall 2697 . . . . . . . . . . 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 205 . . . . . . . . 9 (({⟨𝑥, 𝐵⟩} ∪ {⟨𝑦, 𝑦 / 𝑥𝐵⟩}) ⊆ ⟨𝐶, 𝐷⟩ → (𝑥𝑦 → ((𝑥𝐴𝑦𝐴) → ∃𝑧 𝐴 = {𝑧})))
3420, 33syl6bi 241 . . . . . . . 8 ( 𝑥𝐴 {⟨𝑥, 𝐵⟩} = ⟨𝐶, 𝐷⟩ → (({⟨𝑥, 𝐵⟩} ∪ {⟨𝑦, 𝑦 / 𝑥𝐵⟩}) ⊆ 𝑥𝐴 {⟨𝑥, 𝐵⟩} → (𝑥𝑦 → ((𝑥𝐴𝑦𝐴) → ∃𝑧 𝐴 = {𝑧}))))
3534com14 93 . . . . . . 7 ((𝑥𝐴𝑦𝐴) → (({⟨𝑥, 𝐵⟩} ∪ {⟨𝑦, 𝑦 / 𝑥𝐵⟩}) ⊆ 𝑥𝐴 {⟨𝑥, 𝐵⟩} → (𝑥𝑦 → ( 𝑥𝐴 {⟨𝑥, 𝐵⟩} = ⟨𝐶, 𝐷⟩ → ∃𝑧 𝐴 = {𝑧}))))
3619, 35syl5bi 230 . . . . . 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 2917 . . . 4 (𝑥𝐴 → (∃𝑦𝐴 𝑥𝑦 → ( 𝑥𝐴 {⟨𝑥, 𝐵⟩} = ⟨𝐶, 𝐷⟩ → ∃𝑧 𝐴 = {𝑧})))
395, 38rexlimi 2910 . . 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 392 . 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 381   ∧ wa 382   = wceq 1474  ∃wex 1694   ∈ wcel 1938   ≠ wne 2684  ∃wrex 2801  Vcvv 3077  ⦋csb 3403   ∪ cun 3442   ⊆ wss 3444  ∅c0 3777  {csn 4028  {cpr 4030  ⟨cop 4034  ∪ ciun 4353 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pr 4732 This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1699  df-sb 1831  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-iun 4355 This theorem is referenced by:  funopsn  40249
 Copyright terms: Public domain W3C validator