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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj941 | Structured version Visualization version GIF version |
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj941.1 | ⊢ 𝐺 = (𝑓 ∪ {〈𝑛, 𝐶〉}) |
Ref | Expression |
---|---|
bnj941 | ⊢ (𝐶 ∈ V → ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝐺 Fn 𝑝)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj941.1 | . . . . 5 ⊢ 𝐺 = (𝑓 ∪ {〈𝑛, 𝐶〉}) | |
2 | opeq2 4540 | . . . . . . 7 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → 〈𝑛, 𝐶〉 = 〈𝑛, if(𝐶 ∈ V, 𝐶, ∅)〉) | |
3 | 2 | sneqd 4328 | . . . . . 6 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → {〈𝑛, 𝐶〉} = {〈𝑛, if(𝐶 ∈ V, 𝐶, ∅)〉}) |
4 | 3 | uneq2d 3918 | . . . . 5 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → (𝑓 ∪ {〈𝑛, 𝐶〉}) = (𝑓 ∪ {〈𝑛, if(𝐶 ∈ V, 𝐶, ∅)〉})) |
5 | 1, 4 | syl5eq 2817 | . . . 4 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → 𝐺 = (𝑓 ∪ {〈𝑛, if(𝐶 ∈ V, 𝐶, ∅)〉})) |
6 | 5 | fneq1d 6121 | . . 3 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → (𝐺 Fn 𝑝 ↔ (𝑓 ∪ {〈𝑛, if(𝐶 ∈ V, 𝐶, ∅)〉}) Fn 𝑝)) |
7 | 6 | imbi2d 329 | . 2 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → (((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝐺 Fn 𝑝) ↔ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → (𝑓 ∪ {〈𝑛, if(𝐶 ∈ V, 𝐶, ∅)〉}) Fn 𝑝))) |
8 | eqid 2771 | . . 3 ⊢ (𝑓 ∪ {〈𝑛, if(𝐶 ∈ V, 𝐶, ∅)〉}) = (𝑓 ∪ {〈𝑛, if(𝐶 ∈ V, 𝐶, ∅)〉}) | |
9 | 0ex 4924 | . . . 4 ⊢ ∅ ∈ V | |
10 | 9 | elimel 4289 | . . 3 ⊢ if(𝐶 ∈ V, 𝐶, ∅) ∈ V |
11 | 8, 10 | bnj927 31174 | . 2 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → (𝑓 ∪ {〈𝑛, if(𝐶 ∈ V, 𝐶, ∅)〉}) Fn 𝑝) |
12 | 7, 11 | dedth 4278 | 1 ⊢ (𝐶 ∈ V → ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝐺 Fn 𝑝)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 ∈ wcel 2145 Vcvv 3351 ∪ cun 3721 ∅c0 4063 ifcif 4225 {csn 4316 〈cop 4322 suc csuc 5868 Fn wfn 6026 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pr 5034 ax-reg 8652 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-br 4787 df-opab 4847 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-suc 5872 df-fun 6033 df-fn 6034 |
This theorem is referenced by: bnj945 31179 bnj910 31353 |
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