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Theorem elmptrab 21624
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 6287 . 2 (𝑌 ∈ (𝐹𝑋) → 𝑋𝐷)
3 simp1 1060 . 2 ((𝑋𝐷𝑌𝐶𝜓) → 𝑋𝐷)
4 csbeq1 3534 . . . . . 6 (𝑧 = 𝑋𝑧 / 𝑥𝐵 = 𝑋 / 𝑥𝐵)
5 dfsbcq 3435 . . . . . 6 (𝑧 = 𝑋 → ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑[𝑋 / 𝑥][𝑤 / 𝑦]𝜑))
64, 5rabeqbidv 3193 . . . . 5 (𝑧 = 𝑋 → {𝑤𝑧 / 𝑥𝐵[𝑧 / 𝑥][𝑤 / 𝑦]𝜑} = {𝑤𝑋 / 𝑥𝐵[𝑋 / 𝑥][𝑤 / 𝑦]𝜑})
7 nfcv 2763 . . . . . . 7 𝑧{𝑦𝐵𝜑}
8 nfsbc1v 3453 . . . . . . . 8 𝑥[𝑧 / 𝑥][𝑤 / 𝑦]𝜑
9 nfcsb1v 3547 . . . . . . . 8 𝑥𝑧 / 𝑥𝐵
108, 9nfrab 3121 . . . . . . 7 𝑥{𝑤𝑧 / 𝑥𝐵[𝑧 / 𝑥][𝑤 / 𝑦]𝜑}
11 csbeq1a 3540 . . . . . . . . 9 (𝑥 = 𝑧𝐵 = 𝑧 / 𝑥𝐵)
12 sbceq1a 3444 . . . . . . . . 9 (𝑥 = 𝑧 → (𝜑[𝑧 / 𝑥]𝜑))
1311, 12rabeqbidv 3193 . . . . . . . 8 (𝑥 = 𝑧 → {𝑦𝐵𝜑} = {𝑦𝑧 / 𝑥𝐵[𝑧 / 𝑥]𝜑})
14 nfcv 2763 . . . . . . . . 9 𝑤𝑧 / 𝑥𝐵
15 nfcv 2763 . . . . . . . . 9 𝑦𝑧 / 𝑥𝐵
16 nfcv 2763 . . . . . . . . . 10 𝑦𝑧
17 nfsbc1v 3453 . . . . . . . . . 10 𝑦[𝑤 / 𝑦]𝜑
1816, 17nfsbc 3455 . . . . . . . . 9 𝑦[𝑧 / 𝑥][𝑤 / 𝑦]𝜑
19 nfv 1842 . . . . . . . . 9 𝑤[𝑧 / 𝑥]𝜑
20 sbceq1a 3444 . . . . . . . . . . 11 (𝑦 = 𝑤 → ([𝑧 / 𝑥]𝜑[𝑤 / 𝑦][𝑧 / 𝑥]𝜑))
2120equcoms 1946 . . . . . . . . . 10 (𝑤 = 𝑦 → ([𝑧 / 𝑥]𝜑[𝑤 / 𝑦][𝑧 / 𝑥]𝜑))
22 sbccom 3507 . . . . . . . . . 10 ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑[𝑤 / 𝑦][𝑧 / 𝑥]𝜑)
2321, 22syl6rbbr 279 . . . . . . . . 9 (𝑤 = 𝑦 → ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑[𝑧 / 𝑥]𝜑))
2414, 15, 18, 19, 23cbvrab 3196 . . . . . . . 8 {𝑤𝑧 / 𝑥𝐵[𝑧 / 𝑥][𝑤 / 𝑦]𝜑} = {𝑦𝑧 / 𝑥𝐵[𝑧 / 𝑥]𝜑}
2513, 24syl6eqr 2673 . . . . . . 7 (𝑥 = 𝑧 → {𝑦𝐵𝜑} = {𝑤𝑧 / 𝑥𝐵[𝑧 / 𝑥][𝑤 / 𝑦]𝜑})
267, 10, 25cbvmpt 4747 . . . . . 6 (𝑥𝐷 ↦ {𝑦𝐵𝜑}) = (𝑧𝐷 ↦ {𝑤𝑧 / 𝑥𝐵[𝑧 / 𝑥][𝑤 / 𝑦]𝜑})
271, 26eqtri 2643 . . . . 5 𝐹 = (𝑧𝐷 ↦ {𝑤𝑧 / 𝑥𝐵[𝑧 / 𝑥][𝑤 / 𝑦]𝜑})
28 nfv 1842 . . . . . . . 8 𝑥 𝑧𝐷
299nfel1 2778 . . . . . . . 8 𝑥𝑧 / 𝑥𝐵𝑉
3028, 29nfim 1824 . . . . . . 7 𝑥(𝑧𝐷𝑧 / 𝑥𝐵𝑉)
31 eleq1 2688 . . . . . . . 8 (𝑥 = 𝑧 → (𝑥𝐷𝑧𝐷))
3211eleq1d 2685 . . . . . . . 8 (𝑥 = 𝑧 → (𝐵𝑉𝑧 / 𝑥𝐵𝑉))
3331, 32imbi12d 334 . . . . . . 7 (𝑥 = 𝑧 → ((𝑥𝐷𝐵𝑉) ↔ (𝑧𝐷𝑧 / 𝑥𝐵𝑉)))
34 elmptrab.ex . . . . . . 7 (𝑥𝐷𝐵𝑉)
3530, 33, 34chvar 2261 . . . . . 6 (𝑧𝐷𝑧 / 𝑥𝐵𝑉)
36 rabexg 4810 . . . . . 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 6284 . . . 4 (𝑋𝐷 → (𝐹𝑋) = {𝑤𝑋 / 𝑥𝐵[𝑋 / 𝑥][𝑤 / 𝑦]𝜑})
3938eleq2d 2686 . . 3 (𝑋𝐷 → (𝑌 ∈ (𝐹𝑋) ↔ 𝑌 ∈ {𝑤𝑋 / 𝑥𝐵[𝑋 / 𝑥][𝑤 / 𝑦]𝜑}))
40 dfsbcq 3435 . . . . . . 7 (𝑤 = 𝑌 → ([𝑤 / 𝑦]𝜑[𝑌 / 𝑦]𝜑))
4140sbcbidv 3488 . . . . . 6 (𝑤 = 𝑌 → ([𝑋 / 𝑥][𝑤 / 𝑦]𝜑[𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
4241elrab 3361 . . . . 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 2764 . . . . . . 7 (𝑋𝐷𝑥𝐶)
45 elmptrab.s2 . . . . . . 7 (𝑥 = 𝑋𝐵 = 𝐶)
4644, 45csbiegf 3555 . . . . . 6 (𝑋𝐷𝑋 / 𝑥𝐵 = 𝐶)
4746eleq2d 2686 . . . . 5 (𝑋𝐷 → (𝑌𝑋 / 𝑥𝐵𝑌𝐶))
4847anbi1d 741 . . . 4 (𝑋𝐷 → ((𝑌𝑋 / 𝑥𝐵[𝑋 / 𝑥][𝑌 / 𝑦]𝜑) ↔ (𝑌𝐶[𝑋 / 𝑥][𝑌 / 𝑦]𝜑)))
49 nfv 1842 . . . . . 6 𝑥𝜓
50 nfv 1842 . . . . . 6 𝑦𝜓
51 nfv 1842 . . . . . 6 𝑥 𝑌𝐶
52 elmptrab.s1 . . . . . 6 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝜑𝜓))
5349, 50, 51, 52sbc2iegf 3502 . . . . 5 ((𝑋𝐷𝑌𝐶) → ([𝑋 / 𝑥][𝑌 / 𝑦]𝜑𝜓))
5453pm5.32da 673 . . . 4 (𝑋𝐷 → ((𝑌𝐶[𝑋 / 𝑥][𝑌 / 𝑦]𝜑) ↔ (𝑌𝐶𝜓)))
5543, 48, 543bitrd 294 . . 3 (𝑋𝐷 → (𝑌 ∈ {𝑤𝑋 / 𝑥𝐵[𝑋 / 𝑥][𝑤 / 𝑦]𝜑} ↔ (𝑌𝐶𝜓)))
56 3anass 1041 . . . 4 ((𝑋𝐷𝑌𝐶𝜓) ↔ (𝑋𝐷 ∧ (𝑌𝐶𝜓)))
5756baibr 945 . . 3 (𝑋𝐷 → ((𝑌𝐶𝜓) ↔ (𝑋𝐷𝑌𝐶𝜓)))
5839, 55, 573bitrd 294 . 2 (𝑋𝐷 → (𝑌 ∈ (𝐹𝑋) ↔ (𝑋𝐷𝑌𝐶𝜓)))
592, 3, 58pm5.21nii 368 1 (𝑌 ∈ (𝐹𝑋) ↔ (𝑋𝐷𝑌𝐶𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1037   = wceq 1482  wcel 1989  {crab 2915  Vcvv 3198  [wsbc 3433  csb 3531  cmpt 4727  cfv 5886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fv 5894
This theorem is referenced by:  elmptrab2OLD  21625  elmptrab2  21626  isfbas  21627
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