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Theorem elmptrab 22436
Description: Membership in a one-parameter class of sets. (Contributed by Stefan O'Rear, 28-Jul-2015.)
Hypotheses
Ref Expression
elmptrab.f 𝐹 = (𝑥𝐷 ↦ {𝑦𝐵𝜑})
elmptrab.s1 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝜑𝜓))
elmptrab.s2 (𝑥 = 𝑋𝐵 = 𝐶)
elmptrab.ex (𝑥𝐷𝐵𝑉)
Assertion
Ref Expression
elmptrab (𝑌 ∈ (𝐹𝑋) ↔ (𝑋𝐷𝑌𝐶𝜓))
Distinct variable groups:   𝑥,𝑦,𝑋   𝑦,𝐵   𝑥,𝐶,𝑦   𝑥,𝐷   𝑥,𝑉,𝑦   𝑥,𝑌,𝑦   𝜓,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐵(𝑥)   𝐷(𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem elmptrab
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elmptrab.f . . 3 𝐹 = (𝑥𝐷 ↦ {𝑦𝐵𝜑})
21mptrcl 6758 . 2 (𝑌 ∈ (𝐹𝑋) → 𝑋𝐷)
3 simp1 1133 . 2 ((𝑋𝐷𝑌𝐶𝜓) → 𝑋𝐷)
4 csbeq1 3834 . . . . . 6 (𝑧 = 𝑋𝑧 / 𝑥𝐵 = 𝑋 / 𝑥𝐵)
5 dfsbcq 3725 . . . . . 6 (𝑧 = 𝑋 → ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑[𝑋 / 𝑥][𝑤 / 𝑦]𝜑))
64, 5rabeqbidv 3436 . . . . 5 (𝑧 = 𝑋 → {𝑤𝑧 / 𝑥𝐵[𝑧 / 𝑥][𝑤 / 𝑦]𝜑} = {𝑤𝑋 / 𝑥𝐵[𝑋 / 𝑥][𝑤 / 𝑦]𝜑})
7 nfcv 2958 . . . . . . 7 𝑧{𝑦𝐵𝜑}
8 nfsbc1v 3743 . . . . . . . 8 𝑥[𝑧 / 𝑥][𝑤 / 𝑦]𝜑
9 nfcsb1v 3855 . . . . . . . 8 𝑥𝑧 / 𝑥𝐵
108, 9nfrabw 3341 . . . . . . 7 𝑥{𝑤𝑧 / 𝑥𝐵[𝑧 / 𝑥][𝑤 / 𝑦]𝜑}
11 csbeq1a 3845 . . . . . . . . 9 (𝑥 = 𝑧𝐵 = 𝑧 / 𝑥𝐵)
12 sbceq1a 3734 . . . . . . . . 9 (𝑥 = 𝑧 → (𝜑[𝑧 / 𝑥]𝜑))
1311, 12rabeqbidv 3436 . . . . . . . 8 (𝑥 = 𝑧 → {𝑦𝐵𝜑} = {𝑦𝑧 / 𝑥𝐵[𝑧 / 𝑥]𝜑})
14 nfcv 2958 . . . . . . . . 9 𝑤𝑧 / 𝑥𝐵
15 nfcv 2958 . . . . . . . . 9 𝑦𝑧 / 𝑥𝐵
16 nfcv 2958 . . . . . . . . . 10 𝑦𝑧
17 nfsbc1v 3743 . . . . . . . . . 10 𝑦[𝑤 / 𝑦]𝜑
1816, 17nfsbcw 3745 . . . . . . . . 9 𝑦[𝑧 / 𝑥][𝑤 / 𝑦]𝜑
19 nfv 1915 . . . . . . . . 9 𝑤[𝑧 / 𝑥]𝜑
20 sbceq1a 3734 . . . . . . . . . . 11 (𝑦 = 𝑤 → ([𝑧 / 𝑥]𝜑[𝑤 / 𝑦][𝑧 / 𝑥]𝜑))
2120equcoms 2027 . . . . . . . . . 10 (𝑤 = 𝑦 → ([𝑧 / 𝑥]𝜑[𝑤 / 𝑦][𝑧 / 𝑥]𝜑))
22 sbccom 3803 . . . . . . . . . 10 ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑[𝑤 / 𝑦][𝑧 / 𝑥]𝜑)
2321, 22syl6rbbr 293 . . . . . . . . 9 (𝑤 = 𝑦 → ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑[𝑧 / 𝑥]𝜑))
2414, 15, 18, 19, 23cbvrabw 3440 . . . . . . . 8 {𝑤𝑧 / 𝑥𝐵[𝑧 / 𝑥][𝑤 / 𝑦]𝜑} = {𝑦𝑧 / 𝑥𝐵[𝑧 / 𝑥]𝜑}
2513, 24eqtr4di 2854 . . . . . . 7 (𝑥 = 𝑧 → {𝑦𝐵𝜑} = {𝑤𝑧 / 𝑥𝐵[𝑧 / 𝑥][𝑤 / 𝑦]𝜑})
267, 10, 25cbvmpt 5134 . . . . . 6 (𝑥𝐷 ↦ {𝑦𝐵𝜑}) = (𝑧𝐷 ↦ {𝑤𝑧 / 𝑥𝐵[𝑧 / 𝑥][𝑤 / 𝑦]𝜑})
271, 26eqtri 2824 . . . . 5 𝐹 = (𝑧𝐷 ↦ {𝑤𝑧 / 𝑥𝐵[𝑧 / 𝑥][𝑤 / 𝑦]𝜑})
28 nfv 1915 . . . . . . . 8 𝑥 𝑧𝐷
299nfel1 2974 . . . . . . . 8 𝑥𝑧 / 𝑥𝐵𝑉
3028, 29nfim 1897 . . . . . . 7 𝑥(𝑧𝐷𝑧 / 𝑥𝐵𝑉)
31 eleq1w 2875 . . . . . . . 8 (𝑥 = 𝑧 → (𝑥𝐷𝑧𝐷))
3211eleq1d 2877 . . . . . . . 8 (𝑥 = 𝑧 → (𝐵𝑉𝑧 / 𝑥𝐵𝑉))
3331, 32imbi12d 348 . . . . . . 7 (𝑥 = 𝑧 → ((𝑥𝐷𝐵𝑉) ↔ (𝑧𝐷𝑧 / 𝑥𝐵𝑉)))
34 elmptrab.ex . . . . . . 7 (𝑥𝐷𝐵𝑉)
3530, 33, 34chvarfv 2241 . . . . . 6 (𝑧𝐷𝑧 / 𝑥𝐵𝑉)
36 rabexg 5201 . . . . . 6 (𝑧 / 𝑥𝐵𝑉 → {𝑤𝑧 / 𝑥𝐵[𝑧 / 𝑥][𝑤 / 𝑦]𝜑} ∈ V)
3735, 36syl 17 . . . . 5 (𝑧𝐷 → {𝑤𝑧 / 𝑥𝐵[𝑧 / 𝑥][𝑤 / 𝑦]𝜑} ∈ V)
386, 27, 37fvmpt3 6753 . . . 4 (𝑋𝐷 → (𝐹𝑋) = {𝑤𝑋 / 𝑥𝐵[𝑋 / 𝑥][𝑤 / 𝑦]𝜑})
3938eleq2d 2878 . . 3 (𝑋𝐷 → (𝑌 ∈ (𝐹𝑋) ↔ 𝑌 ∈ {𝑤𝑋 / 𝑥𝐵[𝑋 / 𝑥][𝑤 / 𝑦]𝜑}))
40 dfsbcq 3725 . . . . . . 7 (𝑤 = 𝑌 → ([𝑤 / 𝑦]𝜑[𝑌 / 𝑦]𝜑))
4140sbcbidv 3777 . . . . . 6 (𝑤 = 𝑌 → ([𝑋 / 𝑥][𝑤 / 𝑦]𝜑[𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
4241elrab 3631 . . . . 5 (𝑌 ∈ {𝑤𝑋 / 𝑥𝐵[𝑋 / 𝑥][𝑤 / 𝑦]𝜑} ↔ (𝑌𝑋 / 𝑥𝐵[𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
4342a1i 11 . . . 4 (𝑋𝐷 → (𝑌 ∈ {𝑤𝑋 / 𝑥𝐵[𝑋 / 𝑥][𝑤 / 𝑦]𝜑} ↔ (𝑌𝑋 / 𝑥𝐵[𝑋 / 𝑥][𝑌 / 𝑦]𝜑)))
44 nfcvd 2959 . . . . . . 7 (𝑋𝐷𝑥𝐶)
45 elmptrab.s2 . . . . . . 7 (𝑥 = 𝑋𝐵 = 𝐶)
4644, 45csbiegf 3864 . . . . . 6 (𝑋𝐷𝑋 / 𝑥𝐵 = 𝐶)
4746eleq2d 2878 . . . . 5 (𝑋𝐷 → (𝑌𝑋 / 𝑥𝐵𝑌𝐶))
4847anbi1d 632 . . . 4 (𝑋𝐷 → ((𝑌𝑋 / 𝑥𝐵[𝑋 / 𝑥][𝑌 / 𝑦]𝜑) ↔ (𝑌𝐶[𝑋 / 𝑥][𝑌 / 𝑦]𝜑)))
49 nfv 1915 . . . . . 6 𝑥𝜓
50 nfv 1915 . . . . . 6 𝑦𝜓
51 nfv 1915 . . . . . 6 𝑥 𝑌𝐶
52 elmptrab.s1 . . . . . 6 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝜑𝜓))
5349, 50, 51, 52sbc2iegf 3798 . . . . 5 ((𝑋𝐷𝑌𝐶) → ([𝑋 / 𝑥][𝑌 / 𝑦]𝜑𝜓))
5453pm5.32da 582 . . . 4 (𝑋𝐷 → ((𝑌𝐶[𝑋 / 𝑥][𝑌 / 𝑦]𝜑) ↔ (𝑌𝐶𝜓)))
5543, 48, 543bitrd 308 . . 3 (𝑋𝐷 → (𝑌 ∈ {𝑤𝑋 / 𝑥𝐵[𝑋 / 𝑥][𝑤 / 𝑦]𝜑} ↔ (𝑌𝐶𝜓)))
56 3anass 1092 . . . 4 ((𝑋𝐷𝑌𝐶𝜓) ↔ (𝑋𝐷 ∧ (𝑌𝐶𝜓)))
5756baibr 540 . . 3 (𝑋𝐷 → ((𝑌𝐶𝜓) ↔ (𝑋𝐷𝑌𝐶𝜓)))
5839, 55, 573bitrd 308 . 2 (𝑋𝐷 → (𝑌 ∈ (𝐹𝑋) ↔ (𝑋𝐷𝑌𝐶𝜓)))
592, 3, 58pm5.21nii 383 1 (𝑌 ∈ (𝐹𝑋) ↔ (𝑋𝐷𝑌𝐶𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  {crab 3113  Vcvv 3444  [wsbc 3723  csb 3831  cmpt 5113  cfv 6328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fv 6336
This theorem is referenced by:  elmptrab2  22437  isfbas  22438
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