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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 488 | . . . 4 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝑓 Fn 𝑛) | |
2 | vex 3402 | . . . . . 6 ⊢ 𝑛 ∈ V | |
3 | bnj927.2 | . . . . . 6 ⊢ 𝐶 ∈ V | |
4 | 2, 3 | fnsn 6416 | . . . . 5 ⊢ {〈𝑛, 𝐶〉} Fn {𝑛} |
5 | 4 | a1i 11 | . . . 4 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → {〈𝑛, 𝐶〉} Fn {𝑛}) |
6 | bnj521 32382 | . . . . 5 ⊢ (𝑛 ∩ {𝑛}) = ∅ | |
7 | 6 | a1i 11 | . . . 4 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → (𝑛 ∩ {𝑛}) = ∅) |
8 | 1, 5, 7 | fnund 6469 | . . 3 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → (𝑓 ∪ {〈𝑛, 𝐶〉}) Fn (𝑛 ∪ {𝑛})) |
9 | bnj927.1 | . . . 4 ⊢ 𝐺 = (𝑓 ∪ {〈𝑛, 𝐶〉}) | |
10 | 9 | fneq1i 6454 | . . 3 ⊢ (𝐺 Fn (𝑛 ∪ {𝑛}) ↔ (𝑓 ∪ {〈𝑛, 𝐶〉}) Fn (𝑛 ∪ {𝑛})) |
11 | 8, 10 | sylibr 237 | . 2 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝐺 Fn (𝑛 ∪ {𝑛})) |
12 | df-suc 6197 | . . . . . 6 ⊢ suc 𝑛 = (𝑛 ∪ {𝑛}) | |
13 | 12 | eqeq2i 2749 | . . . . 5 ⊢ (𝑝 = suc 𝑛 ↔ 𝑝 = (𝑛 ∪ {𝑛})) |
14 | 13 | biimpi 219 | . . . 4 ⊢ (𝑝 = suc 𝑛 → 𝑝 = (𝑛 ∪ {𝑛})) |
15 | 14 | adantr 484 | . . 3 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝑝 = (𝑛 ∪ {𝑛})) |
16 | 15 | fneq2d 6451 | . 2 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → (𝐺 Fn 𝑝 ↔ 𝐺 Fn (𝑛 ∪ {𝑛}))) |
17 | 11, 16 | mpbird 260 | 1 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝐺 Fn 𝑝) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 Vcvv 3398 ∪ cun 3851 ∩ cin 3852 ∅c0 4223 {csn 4527 〈cop 4533 suc csuc 6193 Fn wfn 6353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-reg 9186 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-suc 6197 df-fun 6360 df-fn 6361 |
This theorem is referenced by: bnj941 32419 bnj929 32583 |
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