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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj124 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj150 32258. 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.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj124.1 | ⊢ 𝐹 = {〈∅, pred(𝑥, 𝐴, 𝑅)〉} |
bnj124.2 | ⊢ (𝜑″ ↔ [𝐹 / 𝑓]𝜑′) |
bnj124.3 | ⊢ (𝜓″ ↔ [𝐹 / 𝑓]𝜓′) |
bnj124.4 | ⊢ (𝜁″ ↔ [𝐹 / 𝑓]𝜁′) |
bnj124.5 | ⊢ (𝜁′ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′))) |
Ref | Expression |
---|---|
bnj124 | ⊢ (𝜁″ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹 Fn 1o ∧ 𝜑″ ∧ 𝜓″))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj124.4 | . 2 ⊢ (𝜁″ ↔ [𝐹 / 𝑓]𝜁′) | |
2 | bnj124.5 | . . . 4 ⊢ (𝜁′ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′))) | |
3 | 2 | sbcbii 3776 | . . 3 ⊢ ([𝐹 / 𝑓]𝜁′ ↔ [𝐹 / 𝑓]((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′))) |
4 | bnj124.1 | . . . . 5 ⊢ 𝐹 = {〈∅, pred(𝑥, 𝐴, 𝑅)〉} | |
5 | 4 | bnj95 32246 | . . . 4 ⊢ 𝐹 ∈ V |
6 | nfv 1915 | . . . . 5 ⊢ Ⅎ𝑓(𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) | |
7 | 6 | sbc19.21g 3792 | . . . 4 ⊢ (𝐹 ∈ V → ([𝐹 / 𝑓]((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → [𝐹 / 𝑓](𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)))) |
8 | 5, 7 | ax-mp 5 | . . 3 ⊢ ([𝐹 / 𝑓]((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → [𝐹 / 𝑓](𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′))) |
9 | fneq1 6414 | . . . . . . . 8 ⊢ (𝑓 = 𝑧 → (𝑓 Fn 1o ↔ 𝑧 Fn 1o)) | |
10 | fneq1 6414 | . . . . . . . 8 ⊢ (𝑧 = 𝐹 → (𝑧 Fn 1o ↔ 𝐹 Fn 1o)) | |
11 | 9, 10 | sbcie2g 3759 | . . . . . . 7 ⊢ (𝐹 ∈ V → ([𝐹 / 𝑓]𝑓 Fn 1o ↔ 𝐹 Fn 1o)) |
12 | 5, 11 | ax-mp 5 | . . . . . 6 ⊢ ([𝐹 / 𝑓]𝑓 Fn 1o ↔ 𝐹 Fn 1o) |
13 | 12 | bicomi 227 | . . . . 5 ⊢ (𝐹 Fn 1o ↔ [𝐹 / 𝑓]𝑓 Fn 1o) |
14 | bnj124.2 | . . . . 5 ⊢ (𝜑″ ↔ [𝐹 / 𝑓]𝜑′) | |
15 | bnj124.3 | . . . . 5 ⊢ (𝜓″ ↔ [𝐹 / 𝑓]𝜓′) | |
16 | 13, 14, 15, 5 | bnj206 32111 | . . . 4 ⊢ ([𝐹 / 𝑓](𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′) ↔ (𝐹 Fn 1o ∧ 𝜑″ ∧ 𝜓″)) |
17 | 16 | imbi2i 339 | . . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → [𝐹 / 𝑓](𝑓 Fn 1o ∧ 𝜑′ ∧ 𝜓′)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹 Fn 1o ∧ 𝜑″ ∧ 𝜓″))) |
18 | 3, 8, 17 | 3bitri 300 | . 2 ⊢ ([𝐹 / 𝑓]𝜁′ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹 Fn 1o ∧ 𝜑″ ∧ 𝜓″))) |
19 | 1, 18 | bitri 278 | 1 ⊢ (𝜁″ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹 Fn 1o ∧ 𝜑″ ∧ 𝜓″))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 Vcvv 3441 [wsbc 3720 ∅c0 4243 {csn 4525 〈cop 4531 Fn wfn 6319 1oc1o 8078 predc-bnj14 32068 FrSe w-bnj15 32072 |
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 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
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-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-fun 6326 df-fn 6327 |
This theorem is referenced by: bnj150 32258 bnj153 32262 |
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