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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj927 | 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 |
---|---|
bnj927.1 | ⊢ 𝐺 = (𝑓 ∪ {〈𝑛, 𝐶〉}) |
bnj927.2 | ⊢ 𝐶 ∈ V |
Ref | Expression |
---|---|
bnj927 | ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝐺 Fn 𝑝) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 484 | . . . 4 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝑓 Fn 𝑛) | |
2 | vex 3482 | . . . . . 6 ⊢ 𝑛 ∈ V | |
3 | bnj927.2 | . . . . . 6 ⊢ 𝐶 ∈ V | |
4 | 2, 3 | fnsn 6626 | . . . . 5 ⊢ {〈𝑛, 𝐶〉} Fn {𝑛} |
5 | 4 | a1i 11 | . . . 4 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → {〈𝑛, 𝐶〉} Fn {𝑛}) |
6 | disjcsn 9642 | . . . . 5 ⊢ (𝑛 ∩ {𝑛}) = ∅ | |
7 | 6 | a1i 11 | . . . 4 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → (𝑛 ∩ {𝑛}) = ∅) |
8 | 1, 5, 7 | fnund 6684 | . . 3 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → (𝑓 ∪ {〈𝑛, 𝐶〉}) Fn (𝑛 ∪ {𝑛})) |
9 | bnj927.1 | . . . 4 ⊢ 𝐺 = (𝑓 ∪ {〈𝑛, 𝐶〉}) | |
10 | 9 | fneq1i 6666 | . . 3 ⊢ (𝐺 Fn (𝑛 ∪ {𝑛}) ↔ (𝑓 ∪ {〈𝑛, 𝐶〉}) Fn (𝑛 ∪ {𝑛})) |
11 | 8, 10 | sylibr 234 | . 2 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝐺 Fn (𝑛 ∪ {𝑛})) |
12 | df-suc 6392 | . . . . . 6 ⊢ suc 𝑛 = (𝑛 ∪ {𝑛}) | |
13 | 12 | eqeq2i 2748 | . . . . 5 ⊢ (𝑝 = suc 𝑛 ↔ 𝑝 = (𝑛 ∪ {𝑛})) |
14 | 13 | biimpi 216 | . . . 4 ⊢ (𝑝 = suc 𝑛 → 𝑝 = (𝑛 ∪ {𝑛})) |
15 | 14 | adantr 480 | . . 3 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝑝 = (𝑛 ∪ {𝑛})) |
16 | 15 | fneq2d 6663 | . 2 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → (𝐺 Fn 𝑝 ↔ 𝐺 Fn (𝑛 ∪ {𝑛}))) |
17 | 11, 16 | mpbird 257 | 1 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝐺 Fn 𝑝) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∪ cun 3961 ∩ cin 3962 ∅c0 4339 {csn 4631 〈cop 4637 suc csuc 6388 Fn wfn 6558 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-reg 9630 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-suc 6392 df-fun 6565 df-fn 6566 |
This theorem is referenced by: bnj941 34765 bnj929 34929 |
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