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Theorem bnj1123 30759
Description: Technical lemma for bnj69 30783. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1123.4 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj1123.3 𝐾 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
bnj1123.1 (𝜂 ↔ ((𝑓𝐾𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ 𝐵))
bnj1123.2 (𝜂′[𝑗 / 𝑖]𝜂)
Assertion
Ref Expression
bnj1123 (𝜂′ ↔ ((𝑓𝐾𝑗 ∈ dom 𝑓) → (𝑓𝑗) ⊆ 𝐵))
Distinct variable groups:   𝐵,𝑖   𝐷,𝑖   𝑓,𝑖   𝑖,𝑗   𝑖,𝑛   𝜑,𝑖
Allowed substitution hints:   𝜑(𝑦,𝑓,𝑗,𝑛)   𝜓(𝑦,𝑓,𝑖,𝑗,𝑛)   𝜂(𝑦,𝑓,𝑖,𝑗,𝑛)   𝐴(𝑦,𝑓,𝑖,𝑗,𝑛)   𝐵(𝑦,𝑓,𝑗,𝑛)   𝐷(𝑦,𝑓,𝑗,𝑛)   𝑅(𝑦,𝑓,𝑖,𝑗,𝑛)   𝐾(𝑦,𝑓,𝑖,𝑗,𝑛)   𝜂′(𝑦,𝑓,𝑖,𝑗,𝑛)

Proof of Theorem bnj1123
StepHypRef Expression
1 bnj1123.2 . 2 (𝜂′[𝑗 / 𝑖]𝜂)
2 bnj1123.1 . . 3 (𝜂 ↔ ((𝑓𝐾𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ 𝐵))
32sbcbii 3473 . 2 ([𝑗 / 𝑖]𝜂[𝑗 / 𝑖]((𝑓𝐾𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ 𝐵))
4 vex 3189 . . 3 𝑗 ∈ V
5 bnj1123.3 . . . . . . . 8 𝐾 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
6 nfcv 2761 . . . . . . . . . 10 𝑖𝐷
7 nfv 1840 . . . . . . . . . . 11 𝑖 𝑓 Fn 𝑛
8 nfv 1840 . . . . . . . . . . 11 𝑖𝜑
9 bnj1123.4 . . . . . . . . . . . . 13 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
109bnj1095 30557 . . . . . . . . . . . 12 (𝜓 → ∀𝑖𝜓)
1110nf5i 2021 . . . . . . . . . . 11 𝑖𝜓
127, 8, 11nf3an 1828 . . . . . . . . . 10 𝑖(𝑓 Fn 𝑛𝜑𝜓)
136, 12nfrex 3001 . . . . . . . . 9 𝑖𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)
1413nfab 2765 . . . . . . . 8 𝑖{𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
155, 14nfcxfr 2759 . . . . . . 7 𝑖𝐾
1615nfcri 2755 . . . . . 6 𝑖 𝑓𝐾
17 nfv 1840 . . . . . 6 𝑖 𝑗 ∈ dom 𝑓
1816, 17nfan 1825 . . . . 5 𝑖(𝑓𝐾𝑗 ∈ dom 𝑓)
19 nfv 1840 . . . . 5 𝑖(𝑓𝑗) ⊆ 𝐵
2018, 19nfim 1822 . . . 4 𝑖((𝑓𝐾𝑗 ∈ dom 𝑓) → (𝑓𝑗) ⊆ 𝐵)
21 eleq1 2686 . . . . . 6 (𝑖 = 𝑗 → (𝑖 ∈ dom 𝑓𝑗 ∈ dom 𝑓))
2221anbi2d 739 . . . . 5 (𝑖 = 𝑗 → ((𝑓𝐾𝑖 ∈ dom 𝑓) ↔ (𝑓𝐾𝑗 ∈ dom 𝑓)))
23 fveq2 6148 . . . . . 6 (𝑖 = 𝑗 → (𝑓𝑖) = (𝑓𝑗))
2423sseq1d 3611 . . . . 5 (𝑖 = 𝑗 → ((𝑓𝑖) ⊆ 𝐵 ↔ (𝑓𝑗) ⊆ 𝐵))
2522, 24imbi12d 334 . . . 4 (𝑖 = 𝑗 → (((𝑓𝐾𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ 𝐵) ↔ ((𝑓𝐾𝑗 ∈ dom 𝑓) → (𝑓𝑗) ⊆ 𝐵)))
2620, 25sbciegf 3449 . . 3 (𝑗 ∈ V → ([𝑗 / 𝑖]((𝑓𝐾𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ 𝐵) ↔ ((𝑓𝐾𝑗 ∈ dom 𝑓) → (𝑓𝑗) ⊆ 𝐵)))
274, 26ax-mp 5 . 2 ([𝑗 / 𝑖]((𝑓𝐾𝑖 ∈ dom 𝑓) → (𝑓𝑖) ⊆ 𝐵) ↔ ((𝑓𝐾𝑗 ∈ dom 𝑓) → (𝑓𝑗) ⊆ 𝐵))
281, 3, 273bitri 286 1 (𝜂′ ↔ ((𝑓𝐾𝑗 ∈ dom 𝑓) → (𝑓𝑗) ⊆ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  {cab 2607  wral 2907  wrex 2908  Vcvv 3186  [wsbc 3417  wss 3555   ciun 4485  dom cdm 5074  suc csuc 5684   Fn wfn 5842  cfv 5847  ωcom 7012   predc-bnj14 30458
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
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-iota 5810  df-fv 5855
This theorem is referenced by:  bnj1030  30760
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