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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 4705 | . . . . . . 7 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → 〈𝑛, 𝐶〉 = 〈𝑛, if(𝐶 ∈ V, 𝐶, ∅)〉) | |
3 | 2 | sneqd 4478 | . . . . . 6 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → {〈𝑛, 𝐶〉} = {〈𝑛, if(𝐶 ∈ V, 𝐶, ∅)〉}) |
4 | 3 | uneq2d 4055 | . . . . 5 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → (𝑓 ∪ {〈𝑛, 𝐶〉}) = (𝑓 ∪ {〈𝑛, if(𝐶 ∈ V, 𝐶, ∅)〉})) |
5 | 1, 4 | syl5eq 2841 | . . . 4 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → 𝐺 = (𝑓 ∪ {〈𝑛, if(𝐶 ∈ V, 𝐶, ∅)〉})) |
6 | 5 | fneq1d 6308 | . . 3 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → (𝐺 Fn 𝑝 ↔ (𝑓 ∪ {〈𝑛, if(𝐶 ∈ V, 𝐶, ∅)〉}) Fn 𝑝)) |
7 | 6 | imbi2d 342 | . 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 2793 | . . 3 ⊢ (𝑓 ∪ {〈𝑛, if(𝐶 ∈ V, 𝐶, ∅)〉}) = (𝑓 ∪ {〈𝑛, if(𝐶 ∈ V, 𝐶, ∅)〉}) | |
9 | 0ex 5096 | . . . 4 ⊢ ∅ ∈ V | |
10 | 9 | elimel 4442 | . . 3 ⊢ if(𝐶 ∈ V, 𝐶, ∅) ∈ V |
11 | 8, 10 | bnj927 31613 | . 2 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → (𝑓 ∪ {〈𝑛, if(𝐶 ∈ V, 𝐶, ∅)〉}) Fn 𝑝) |
12 | 7, 11 | dedth 4431 | 1 ⊢ (𝐶 ∈ V → ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝐺 Fn 𝑝)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1520 ∈ wcel 2079 Vcvv 3432 ∪ cun 3852 ∅c0 4206 ifcif 4375 {csn 4466 〈cop 4472 suc csuc 6060 Fn wfn 6212 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-sep 5088 ax-nul 5095 ax-pr 5214 ax-reg 8892 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-ral 3108 df-rex 3109 df-rab 3112 df-v 3434 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-nul 4207 df-if 4376 df-sn 4467 df-pr 4469 df-op 4473 df-br 4957 df-opab 5019 df-id 5340 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-suc 6064 df-fun 6219 df-fn 6220 |
This theorem is referenced by: bnj945 31618 bnj910 31792 |
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