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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj97 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj150 32148. 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.) |
Ref | Expression |
---|---|
bnj96.1 | ⊢ 𝐹 = {〈∅, pred(𝑥, 𝐴, 𝑅)〉} |
Ref | Expression |
---|---|
bnj97 | ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj93 32135 | . . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V) | |
2 | 0ex 5211 | . . . . 5 ⊢ ∅ ∈ V | |
3 | 2 | bnj519 32006 | . . . 4 ⊢ ( pred(𝑥, 𝐴, 𝑅) ∈ V → Fun {〈∅, pred(𝑥, 𝐴, 𝑅)〉}) |
4 | bnj96.1 | . . . . 5 ⊢ 𝐹 = {〈∅, pred(𝑥, 𝐴, 𝑅)〉} | |
5 | 4 | funeqi 6376 | . . . 4 ⊢ (Fun 𝐹 ↔ Fun {〈∅, pred(𝑥, 𝐴, 𝑅)〉}) |
6 | 3, 5 | sylibr 236 | . . 3 ⊢ ( pred(𝑥, 𝐴, 𝑅) ∈ V → Fun 𝐹) |
7 | 1, 6 | syl 17 | . 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → Fun 𝐹) |
8 | opex 5356 | . . . 4 ⊢ 〈∅, pred(𝑥, 𝐴, 𝑅)〉 ∈ V | |
9 | 8 | snid 4601 | . . 3 ⊢ 〈∅, pred(𝑥, 𝐴, 𝑅)〉 ∈ {〈∅, pred(𝑥, 𝐴, 𝑅)〉} |
10 | 9, 4 | eleqtrri 2912 | . 2 ⊢ 〈∅, pred(𝑥, 𝐴, 𝑅)〉 ∈ 𝐹 |
11 | funopfv 6717 | . 2 ⊢ (Fun 𝐹 → (〈∅, pred(𝑥, 𝐴, 𝑅)〉 ∈ 𝐹 → (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))) | |
12 | 7, 10, 11 | mpisyl 21 | 1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∅c0 4291 {csn 4567 〈cop 4573 Fun wfun 6349 ‘cfv 6355 predc-bnj14 31958 FrSe w-bnj15 31962 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fv 6363 df-bnj13 31961 df-bnj15 31963 |
This theorem is referenced by: bnj150 32148 |
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