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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 4341 | . . . . . . 7 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → 〈𝑛, 𝐶〉 = 〈𝑛, if(𝐶 ∈ V, 𝐶, ∅)〉) | |
3 | 2 | sneqd 4137 | . . . . . 6 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → {〈𝑛, 𝐶〉} = {〈𝑛, if(𝐶 ∈ V, 𝐶, ∅)〉}) |
4 | 3 | uneq2d 3729 | . . . . 5 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → (𝑓 ∪ {〈𝑛, 𝐶〉}) = (𝑓 ∪ {〈𝑛, if(𝐶 ∈ V, 𝐶, ∅)〉})) |
5 | 1, 4 | syl5eq 2656 | . . . 4 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → 𝐺 = (𝑓 ∪ {〈𝑛, if(𝐶 ∈ V, 𝐶, ∅)〉})) |
6 | 5 | fneq1d 5895 | . . 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 2610 | . . 3 ⊢ (𝑓 ∪ {〈𝑛, if(𝐶 ∈ V, 𝐶, ∅)〉}) = (𝑓 ∪ {〈𝑛, if(𝐶 ∈ V, 𝐶, ∅)〉}) | |
9 | 0ex 4718 | . . . 4 ⊢ ∅ ∈ V | |
10 | 9 | elimel 4100 | . . 3 ⊢ if(𝐶 ∈ V, 𝐶, ∅) ∈ V |
11 | 8, 10 | bnj927 30093 | . 2 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → (𝑓 ∪ {〈𝑛, if(𝐶 ∈ V, 𝐶, ∅)〉}) Fn 𝑝) |
12 | 7, 11 | dedth 4089 | 1 ⊢ (𝐶 ∈ V → ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝐺 Fn 𝑝)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∪ cun 3538 ∅c0 3874 ifcif 4036 {csn 4125 〈cop 4131 suc csuc 5642 Fn wfn 5799 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-reg 8380 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-suc 5646 df-fun 5806 df-fn 5807 |
This theorem is referenced by: bnj945 30098 bnj910 30272 |
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