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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 4874 | . . . . . . 7 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → 〈𝑛, 𝐶〉 = 〈𝑛, if(𝐶 ∈ V, 𝐶, ∅)〉) | |
| 3 | 2 | sneqd 4638 | . . . . . 6 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → {〈𝑛, 𝐶〉} = {〈𝑛, if(𝐶 ∈ V, 𝐶, ∅)〉}) |
| 4 | 3 | uneq2d 4168 | . . . . 5 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → (𝑓 ∪ {〈𝑛, 𝐶〉}) = (𝑓 ∪ {〈𝑛, if(𝐶 ∈ V, 𝐶, ∅)〉})) |
| 5 | 1, 4 | eqtrid 2789 | . . . 4 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → 𝐺 = (𝑓 ∪ {〈𝑛, if(𝐶 ∈ V, 𝐶, ∅)〉})) |
| 6 | 5 | fneq1d 6661 | . . 3 ⊢ (𝐶 = if(𝐶 ∈ V, 𝐶, ∅) → (𝐺 Fn 𝑝 ↔ (𝑓 ∪ {〈𝑛, if(𝐶 ∈ V, 𝐶, ∅)〉}) Fn 𝑝)) |
| 7 | 6 | imbi2d 340 | . 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 2737 | . . 3 ⊢ (𝑓 ∪ {〈𝑛, if(𝐶 ∈ V, 𝐶, ∅)〉}) = (𝑓 ∪ {〈𝑛, if(𝐶 ∈ V, 𝐶, ∅)〉}) | |
| 9 | 0ex 5307 | . . . 4 ⊢ ∅ ∈ V | |
| 10 | 9 | elimel 4595 | . . 3 ⊢ if(𝐶 ∈ V, 𝐶, ∅) ∈ V |
| 11 | 8, 10 | bnj927 34783 | . 2 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → (𝑓 ∪ {〈𝑛, if(𝐶 ∈ V, 𝐶, ∅)〉}) Fn 𝑝) |
| 12 | 7, 11 | dedth 4584 | 1 ⊢ (𝐶 ∈ V → ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝐺 Fn 𝑝)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∪ cun 3949 ∅c0 4333 ifcif 4525 {csn 4626 〈cop 4632 suc csuc 6386 Fn wfn 6556 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-reg 9632 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-suc 6390 df-fun 6563 df-fn 6564 |
| This theorem is referenced by: bnj945 34787 bnj910 34962 |
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