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 477 | . . . 4 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝑓 Fn 𝑛) | |
2 | vex 3203 | . . . . . 6 ⊢ 𝑛 ∈ V | |
3 | bnj927.2 | . . . . . 6 ⊢ 𝐶 ∈ V | |
4 | 2, 3 | fnsn 5946 | . . . . 5 ⊢ {〈𝑛, 𝐶〉} Fn {𝑛} |
5 | 4 | a1i 11 | . . . 4 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → {〈𝑛, 𝐶〉} Fn {𝑛}) |
6 | bnj521 30805 | . . . . 5 ⊢ (𝑛 ∩ {𝑛}) = ∅ | |
7 | 6 | a1i 11 | . . . 4 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → (𝑛 ∩ {𝑛}) = ∅) |
8 | fnun 5997 | . . . 4 ⊢ (((𝑓 Fn 𝑛 ∧ {〈𝑛, 𝐶〉} Fn {𝑛}) ∧ (𝑛 ∩ {𝑛}) = ∅) → (𝑓 ∪ {〈𝑛, 𝐶〉}) Fn (𝑛 ∪ {𝑛})) | |
9 | 1, 5, 7, 8 | syl21anc 1325 | . . 3 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → (𝑓 ∪ {〈𝑛, 𝐶〉}) Fn (𝑛 ∪ {𝑛})) |
10 | bnj927.1 | . . . 4 ⊢ 𝐺 = (𝑓 ∪ {〈𝑛, 𝐶〉}) | |
11 | 10 | fneq1i 5985 | . . 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 5729 | . . . . . 6 ⊢ suc 𝑛 = (𝑛 ∪ {𝑛}) | |
14 | 13 | eqeq2i 2634 | . . . . 5 ⊢ (𝑝 = suc 𝑛 ↔ 𝑝 = (𝑛 ∪ {𝑛})) |
15 | 14 | biimpi 206 | . . . 4 ⊢ (𝑝 = suc 𝑛 → 𝑝 = (𝑛 ∪ {𝑛})) |
16 | 15 | adantr 481 | . . 3 ⊢ ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝑝 = (𝑛 ∪ {𝑛})) |
17 | 16 | fneq2d 5982 | . 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 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∪ cun 3572 ∩ cin 3573 ∅c0 3915 {csn 4177 〈cop 4183 suc csuc 5725 Fn wfn 5883 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-reg 8497 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-suc 5729 df-fun 5890 df-fn 5891 |
This theorem is referenced by: bnj941 30843 bnj929 31006 |
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