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Theorem opiota 7378
Description: The property of a uniquely specified ordered pair. The proof uses properties of the description binder. (Contributed by Mario Carneiro, 21-May-2015.)
Hypotheses
Ref Expression
opiota.1 𝐼 = (℩𝑧𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
opiota.2 𝑋 = (1st𝐼)
opiota.3 𝑌 = (2nd𝐼)
opiota.4 (𝑥 = 𝐶 → (𝜑𝜓))
opiota.5 (𝑦 = 𝐷 → (𝜓𝜒))
Assertion
Ref Expression
opiota (∃!𝑧𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ((𝐶𝐴𝐷𝐵𝜒) ↔ (𝐶 = 𝑋𝐷 = 𝑌)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝜒,𝑦   𝜑,𝑧   𝑥,𝐷,𝑦,𝑧   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦,𝑧)   𝜒(𝑥,𝑧)   𝐼(𝑥,𝑦,𝑧)   𝑋(𝑥,𝑦,𝑧)   𝑌(𝑥,𝑦,𝑧)

Proof of Theorem opiota
StepHypRef Expression
1 opiota.4 . . . . . . 7 (𝑥 = 𝐶 → (𝜑𝜓))
2 opiota.5 . . . . . . 7 (𝑦 = 𝐷 → (𝜓𝜒))
31, 2ceqsrex2v 3488 . . . . . 6 ((𝐶𝐴𝐷𝐵) → (∃𝑥𝐴𝑦𝐵 ((𝑥 = 𝐶𝑦 = 𝐷) ∧ 𝜑) ↔ 𝜒))
43bicomd 213 . . . . 5 ((𝐶𝐴𝐷𝐵) → (𝜒 ↔ ∃𝑥𝐴𝑦𝐵 ((𝑥 = 𝐶𝑦 = 𝐷) ∧ 𝜑)))
5 opex 5060 . . . . . . . 8 𝐶, 𝐷⟩ ∈ V
65a1i 11 . . . . . . 7 (∃!𝑧𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ⟨𝐶, 𝐷⟩ ∈ V)
7 id 22 . . . . . . 7 (∃!𝑧𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ∃!𝑧𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
8 eqeq1 2775 . . . . . . . . . . 11 (𝑧 = ⟨𝐶, 𝐷⟩ → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ ⟨𝐶, 𝐷⟩ = ⟨𝑥, 𝑦⟩))
9 eqcom 2778 . . . . . . . . . . . 12 (⟨𝐶, 𝐷⟩ = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑥, 𝑦⟩ = ⟨𝐶, 𝐷⟩)
10 vex 3354 . . . . . . . . . . . . 13 𝑥 ∈ V
11 vex 3354 . . . . . . . . . . . . 13 𝑦 ∈ V
1210, 11opth 5072 . . . . . . . . . . . 12 (⟨𝑥, 𝑦⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝑥 = 𝐶𝑦 = 𝐷))
139, 12bitri 264 . . . . . . . . . . 11 (⟨𝐶, 𝐷⟩ = ⟨𝑥, 𝑦⟩ ↔ (𝑥 = 𝐶𝑦 = 𝐷))
148, 13syl6bb 276 . . . . . . . . . 10 (𝑧 = ⟨𝐶, 𝐷⟩ → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ (𝑥 = 𝐶𝑦 = 𝐷)))
1514anbi1d 615 . . . . . . . . 9 (𝑧 = ⟨𝐶, 𝐷⟩ → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ((𝑥 = 𝐶𝑦 = 𝐷) ∧ 𝜑)))
16152rexbidv 3205 . . . . . . . 8 (𝑧 = ⟨𝐶, 𝐷⟩ → (∃𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥𝐴𝑦𝐵 ((𝑥 = 𝐶𝑦 = 𝐷) ∧ 𝜑)))
1716adantl 467 . . . . . . 7 ((∃!𝑧𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∧ 𝑧 = ⟨𝐶, 𝐷⟩) → (∃𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥𝐴𝑦𝐵 ((𝑥 = 𝐶𝑦 = 𝐷) ∧ 𝜑)))
18 nfeu1 2628 . . . . . . 7 𝑧∃!𝑧𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
19 nfvd 1996 . . . . . . 7 (∃!𝑧𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → Ⅎ𝑧𝑥𝐴𝑦𝐵 ((𝑥 = 𝐶𝑦 = 𝐷) ∧ 𝜑))
20 nfcvd 2914 . . . . . . 7 (∃!𝑧𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝑧𝐶, 𝐷⟩)
216, 7, 17, 18, 19, 20iota2df 6018 . . . . . 6 (∃!𝑧𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → (∃𝑥𝐴𝑦𝐵 ((𝑥 = 𝐶𝑦 = 𝐷) ∧ 𝜑) ↔ (℩𝑧𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) = ⟨𝐶, 𝐷⟩))
22 eqcom 2778 . . . . . . 7 (⟨𝐶, 𝐷⟩ = 𝐼𝐼 = ⟨𝐶, 𝐷⟩)
23 opiota.1 . . . . . . . 8 𝐼 = (℩𝑧𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
2423eqeq1i 2776 . . . . . . 7 (𝐼 = ⟨𝐶, 𝐷⟩ ↔ (℩𝑧𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) = ⟨𝐶, 𝐷⟩)
2522, 24bitri 264 . . . . . 6 (⟨𝐶, 𝐷⟩ = 𝐼 ↔ (℩𝑧𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) = ⟨𝐶, 𝐷⟩)
2621, 25syl6bbr 278 . . . . 5 (∃!𝑧𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → (∃𝑥𝐴𝑦𝐵 ((𝑥 = 𝐶𝑦 = 𝐷) ∧ 𝜑) ↔ ⟨𝐶, 𝐷⟩ = 𝐼))
274, 26sylan9bbr 500 . . . 4 ((∃!𝑧𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∧ (𝐶𝐴𝐷𝐵)) → (𝜒 ↔ ⟨𝐶, 𝐷⟩ = 𝐼))
2827pm5.32da 568 . . 3 (∃!𝑧𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → (((𝐶𝐴𝐷𝐵) ∧ 𝜒) ↔ ((𝐶𝐴𝐷𝐵) ∧ ⟨𝐶, 𝐷⟩ = 𝐼)))
29 opelxpi 5288 . . . . . . . . . 10 ((𝑥𝐴𝑦𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵))
30 simpl 468 . . . . . . . . . . 11 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝑧 = ⟨𝑥, 𝑦⟩)
3130eleq1d 2835 . . . . . . . . . 10 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → (𝑧 ∈ (𝐴 × 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
3229, 31syl5ibrcom 237 . . . . . . . . 9 ((𝑥𝐴𝑦𝐵) → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝑧 ∈ (𝐴 × 𝐵)))
3332rexlimivv 3184 . . . . . . . 8 (∃𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝑧 ∈ (𝐴 × 𝐵))
3433abssi 3826 . . . . . . 7 {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ⊆ (𝐴 × 𝐵)
35 iotacl 6017 . . . . . . 7 (∃!𝑧𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → (℩𝑧𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ∈ {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)})
3634, 35sseldi 3750 . . . . . 6 (∃!𝑧𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → (℩𝑧𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ∈ (𝐴 × 𝐵))
3723, 36syl5eqel 2854 . . . . 5 (∃!𝑧𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝐼 ∈ (𝐴 × 𝐵))
38 opelxp 5286 . . . . . 6 (⟨𝐶, 𝐷⟩ ∈ (𝐴 × 𝐵) ↔ (𝐶𝐴𝐷𝐵))
39 eleq1 2838 . . . . . 6 (⟨𝐶, 𝐷⟩ = 𝐼 → (⟨𝐶, 𝐷⟩ ∈ (𝐴 × 𝐵) ↔ 𝐼 ∈ (𝐴 × 𝐵)))
4038, 39syl5bbr 274 . . . . 5 (⟨𝐶, 𝐷⟩ = 𝐼 → ((𝐶𝐴𝐷𝐵) ↔ 𝐼 ∈ (𝐴 × 𝐵)))
4137, 40syl5ibrcom 237 . . . 4 (∃!𝑧𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → (⟨𝐶, 𝐷⟩ = 𝐼 → (𝐶𝐴𝐷𝐵)))
4241pm4.71rd 552 . . 3 (∃!𝑧𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → (⟨𝐶, 𝐷⟩ = 𝐼 ↔ ((𝐶𝐴𝐷𝐵) ∧ ⟨𝐶, 𝐷⟩ = 𝐼)))
43 1st2nd2 7354 . . . . 5 (𝐼 ∈ (𝐴 × 𝐵) → 𝐼 = ⟨(1st𝐼), (2nd𝐼)⟩)
4437, 43syl 17 . . . 4 (∃!𝑧𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝐼 = ⟨(1st𝐼), (2nd𝐼)⟩)
4544eqeq2d 2781 . . 3 (∃!𝑧𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → (⟨𝐶, 𝐷⟩ = 𝐼 ↔ ⟨𝐶, 𝐷⟩ = ⟨(1st𝐼), (2nd𝐼)⟩))
4628, 42, 453bitr2d 296 . 2 (∃!𝑧𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → (((𝐶𝐴𝐷𝐵) ∧ 𝜒) ↔ ⟨𝐶, 𝐷⟩ = ⟨(1st𝐼), (2nd𝐼)⟩))
47 df-3an 1073 . 2 ((𝐶𝐴𝐷𝐵𝜒) ↔ ((𝐶𝐴𝐷𝐵) ∧ 𝜒))
48 opiota.2 . . . . 5 𝑋 = (1st𝐼)
4948eqeq2i 2783 . . . 4 (𝐶 = 𝑋𝐶 = (1st𝐼))
50 opiota.3 . . . . 5 𝑌 = (2nd𝐼)
5150eqeq2i 2783 . . . 4 (𝐷 = 𝑌𝐷 = (2nd𝐼))
5249, 51anbi12i 612 . . 3 ((𝐶 = 𝑋𝐷 = 𝑌) ↔ (𝐶 = (1st𝐼) ∧ 𝐷 = (2nd𝐼)))
53 fvex 6342 . . . 4 (1st𝐼) ∈ V
54 fvex 6342 . . . 4 (2nd𝐼) ∈ V
5553, 54opth2 5076 . . 3 (⟨𝐶, 𝐷⟩ = ⟨(1st𝐼), (2nd𝐼)⟩ ↔ (𝐶 = (1st𝐼) ∧ 𝐷 = (2nd𝐼)))
5652, 55bitr4i 267 . 2 ((𝐶 = 𝑋𝐷 = 𝑌) ↔ ⟨𝐶, 𝐷⟩ = ⟨(1st𝐼), (2nd𝐼)⟩)
5746, 47, 563bitr4g 303 1 (∃!𝑧𝑥𝐴𝑦𝐵 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ((𝐶𝐴𝐷𝐵𝜒) ↔ (𝐶 = 𝑋𝐷 = 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  ∃!weu 2618  {cab 2757  wrex 3062  Vcvv 3351  cop 4322   × cxp 5247  cio 5992  cfv 6031  1st c1st 7313  2nd c2nd 7314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fv 6039  df-1st 7315  df-2nd 7316
This theorem is referenced by:  oeeui  7836
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