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Mirrors > Home > MPE Home > Th. List > fnunop | Structured version Visualization version GIF version |
Description: Extension of a function with a new ordered pair. (Contributed by NM, 28-Sep-2013.) (Revised by Mario Carneiro, 30-Apr-2015.) (Revised by AV, 16-Aug-2024.) |
Ref | Expression |
---|---|
fnunop.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
fnunop.y | ⊢ (𝜑 → 𝑌 ∈ 𝑊) |
fnunop.f | ⊢ (𝜑 → 𝐹 Fn 𝐷) |
fnunop.g | ⊢ 𝐺 = (𝐹 ∪ {⟨𝑋, 𝑌⟩}) |
fnunop.e | ⊢ 𝐸 = (𝐷 ∪ {𝑋}) |
fnunop.d | ⊢ (𝜑 → ¬ 𝑋 ∈ 𝐷) |
Ref | Expression |
---|---|
fnunop | ⊢ (𝜑 → 𝐺 Fn 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnunop.f | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐷) | |
2 | fnunop.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
3 | fnunop.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑊) | |
4 | fnsng 6554 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → {⟨𝑋, 𝑌⟩} Fn {𝑋}) | |
5 | 2, 3, 4 | syl2anc 585 | . . 3 ⊢ (𝜑 → {⟨𝑋, 𝑌⟩} Fn {𝑋}) |
6 | fnunop.d | . . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝐷) | |
7 | disjsn 4673 | . . . 4 ⊢ ((𝐷 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ 𝐷) | |
8 | 6, 7 | sylibr 233 | . . 3 ⊢ (𝜑 → (𝐷 ∩ {𝑋}) = ∅) |
9 | 1, 5, 8 | fnund 6616 | . 2 ⊢ (𝜑 → (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn (𝐷 ∪ {𝑋})) |
10 | fnunop.g | . . . 4 ⊢ 𝐺 = (𝐹 ∪ {⟨𝑋, 𝑌⟩}) | |
11 | 10 | fneq1i 6600 | . . 3 ⊢ (𝐺 Fn 𝐸 ↔ (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn 𝐸) |
12 | fnunop.e | . . . 4 ⊢ 𝐸 = (𝐷 ∪ {𝑋}) | |
13 | 12 | fneq2i 6601 | . . 3 ⊢ ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn 𝐸 ↔ (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn (𝐷 ∪ {𝑋})) |
14 | 11, 13 | bitri 275 | . 2 ⊢ (𝐺 Fn 𝐸 ↔ (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn (𝐷 ∪ {𝑋})) |
15 | 9, 14 | sylibr 233 | 1 ⊢ (𝜑 → 𝐺 Fn 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1542 ∈ wcel 2107 ∪ cun 3909 ∩ cin 3910 ∅c0 4283 {csn 4587 ⟨cop 4593 Fn wfn 6492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-fun 6499 df-fn 6500 |
This theorem is referenced by: fineqvac 33755 |
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