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 485 | . . . 4 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝑓 Fn 𝑛) | |
2 | vex 3498 | . . . . . 6 ⊢ 𝑛 ∈ V | |
3 | bnj927.2 | . . . . . 6 ⊢ 𝐶 ∈ V | |
4 | 2, 3 | fnsn 6406 | . . . . 5 ⊢ {〈𝑛, 𝐶〉} Fn {𝑛} |
5 | 4 | a1i 11 | . . . 4 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → {〈𝑛, 𝐶〉} Fn {𝑛}) |
6 | bnj521 31907 | . . . . 5 ⊢ (𝑛 ∩ {𝑛}) = ∅ | |
7 | 6 | a1i 11 | . . . 4 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → (𝑛 ∩ {𝑛}) = ∅) |
8 | fnun 6457 | . . . 4 ⊢ (((𝑓 Fn 𝑛 ∧ {〈𝑛, 𝐶〉} Fn {𝑛}) ∧ (𝑛 ∩ {𝑛}) = ∅) → (𝑓 ∪ {〈𝑛, 𝐶〉}) Fn (𝑛 ∪ {𝑛})) | |
9 | 1, 5, 7, 8 | syl21anc 833 | . . 3 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → (𝑓 ∪ {〈𝑛, 𝐶〉}) Fn (𝑛 ∪ {𝑛})) |
10 | bnj927.1 | . . . 4 ⊢ 𝐺 = (𝑓 ∪ {〈𝑛, 𝐶〉}) | |
11 | 10 | fneq1i 6444 | . . 3 ⊢ (𝐺 Fn (𝑛 ∪ {𝑛}) ↔ (𝑓 ∪ {〈𝑛, 𝐶〉}) Fn (𝑛 ∪ {𝑛})) |
12 | 9, 11 | sylibr 235 | . 2 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝐺 Fn (𝑛 ∪ {𝑛})) |
13 | df-suc 6191 | . . . . . 6 ⊢ suc 𝑛 = (𝑛 ∪ {𝑛}) | |
14 | 13 | eqeq2i 2834 | . . . . 5 ⊢ (𝑝 = suc 𝑛 ↔ 𝑝 = (𝑛 ∪ {𝑛})) |
15 | 14 | biimpi 217 | . . . 4 ⊢ (𝑝 = suc 𝑛 → 𝑝 = (𝑛 ∪ {𝑛})) |
16 | 15 | adantr 481 | . . 3 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝑝 = (𝑛 ∪ {𝑛})) |
17 | 16 | fneq2d 6441 | . 2 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → (𝐺 Fn 𝑝 ↔ 𝐺 Fn (𝑛 ∪ {𝑛}))) |
18 | 12, 17 | mpbird 258 | 1 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝐺 Fn 𝑝) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3495 ∪ cun 3933 ∩ cin 3934 ∅c0 4290 {csn 4559 〈cop 4565 suc csuc 6187 Fn wfn 6344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 ax-reg 9045 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 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 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-suc 6191 df-fun 6351 df-fn 6352 |
This theorem is referenced by: bnj941 31944 bnj929 32108 |
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