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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 4798 | . . . . . . 7 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → 〈𝑛, 𝐶〉 = 〈𝑛, if(𝐶 ∈ V, 𝐶, ∅)〉) | |
3 | 2 | sneqd 4573 | . . . . . 6 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → {〈𝑛, 𝐶〉} = {〈𝑛, if(𝐶 ∈ V, 𝐶, ∅)〉}) |
4 | 3 | uneq2d 4139 | . . . . 5 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → (𝑓 ∪ {〈𝑛, 𝐶〉}) = (𝑓 ∪ {〈𝑛, if(𝐶 ∈ V, 𝐶, ∅)〉})) |
5 | 1, 4 | syl5eq 2868 | . . . 4 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → 𝐺 = (𝑓 ∪ {〈𝑛, if(𝐶 ∈ V, 𝐶, ∅)〉})) |
6 | 5 | fneq1d 6441 | . . 3 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → (𝐺 Fn 𝑝 ↔ (𝑓 ∪ {〈𝑛, if(𝐶 ∈ V, 𝐶, ∅)〉}) Fn 𝑝)) |
7 | 6 | imbi2d 343 | . 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 2821 | . . 3 ⊢ (𝑓 ∪ {〈𝑛, if(𝐶 ∈ V, 𝐶, ∅)〉}) = (𝑓 ∪ {〈𝑛, if(𝐶 ∈ V, 𝐶, ∅)〉}) | |
9 | 0ex 5204 | . . . 4 ⊢ ∅ ∈ V | |
10 | 9 | elimel 4534 | . . 3 ⊢ if(𝐶 ∈ V, 𝐶, ∅) ∈ V |
11 | 8, 10 | bnj927 32035 | . 2 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → (𝑓 ∪ {〈𝑛, if(𝐶 ∈ V, 𝐶, ∅)〉}) Fn 𝑝) |
12 | 7, 11 | dedth 4523 | 1 ⊢ (𝐶 ∈ V → ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝐺 Fn 𝑝)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3495 ∪ cun 3934 ∅c0 4291 ifcif 4467 {csn 4561 〈cop 4567 suc csuc 6188 Fn wfn 6345 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 ax-reg 9050 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-br 5060 df-opab 5122 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-suc 6192 df-fun 6352 df-fn 6353 |
This theorem is referenced by: bnj945 32040 bnj910 32215 |
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