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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 476 | . . . 4 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝑓 Fn 𝑛) | |
2 | vex 3234 | . . . . . 6 ⊢ 𝑛 ∈ V | |
3 | bnj927.2 | . . . . . 6 ⊢ 𝐶 ∈ V | |
4 | 2, 3 | fnsn 5984 | . . . . 5 ⊢ {〈𝑛, 𝐶〉} Fn {𝑛} |
5 | 4 | a1i 11 | . . . 4 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → {〈𝑛, 𝐶〉} Fn {𝑛}) |
6 | bnj521 30931 | . . . . 5 ⊢ (𝑛 ∩ {𝑛}) = ∅ | |
7 | 6 | a1i 11 | . . . 4 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → (𝑛 ∩ {𝑛}) = ∅) |
8 | fnun 6035 | . . . 4 ⊢ (((𝑓 Fn 𝑛 ∧ {〈𝑛, 𝐶〉} Fn {𝑛}) ∧ (𝑛 ∩ {𝑛}) = ∅) → (𝑓 ∪ {〈𝑛, 𝐶〉}) Fn (𝑛 ∪ {𝑛})) | |
9 | 1, 5, 7, 8 | syl21anc 1365 | . . 3 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → (𝑓 ∪ {〈𝑛, 𝐶〉}) Fn (𝑛 ∪ {𝑛})) |
10 | bnj927.1 | . . . 4 ⊢ 𝐺 = (𝑓 ∪ {〈𝑛, 𝐶〉}) | |
11 | 10 | fneq1i 6023 | . . 3 ⊢ (𝐺 Fn (𝑛 ∪ {𝑛}) ↔ (𝑓 ∪ {〈𝑛, 𝐶〉}) Fn (𝑛 ∪ {𝑛})) |
12 | 9, 11 | sylibr 224 | . 2 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝐺 Fn (𝑛 ∪ {𝑛})) |
13 | df-suc 5767 | . . . . . 6 ⊢ suc 𝑛 = (𝑛 ∪ {𝑛}) | |
14 | 13 | eqeq2i 2663 | . . . . 5 ⊢ (𝑝 = suc 𝑛 ↔ 𝑝 = (𝑛 ∪ {𝑛})) |
15 | 14 | biimpi 206 | . . . 4 ⊢ (𝑝 = suc 𝑛 → 𝑝 = (𝑛 ∪ {𝑛})) |
16 | 15 | adantr 480 | . . 3 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝑝 = (𝑛 ∪ {𝑛})) |
17 | 16 | fneq2d 6020 | . 2 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → (𝐺 Fn 𝑝 ↔ 𝐺 Fn (𝑛 ∪ {𝑛}))) |
18 | 12, 17 | mpbird 247 | 1 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝐺 Fn 𝑝) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 Vcvv 3231 ∪ cun 3605 ∩ cin 3606 ∅c0 3948 {csn 4210 〈cop 4216 suc csuc 5763 Fn wfn 5921 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936 ax-reg 8538 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-br 4686 df-opab 4746 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-suc 5767 df-fun 5928 df-fn 5929 |
This theorem is referenced by: bnj941 30969 bnj929 31132 |
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