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Mirrors > Home > MPE Home > Th. List > rnmptcOLD | Structured version Visualization version GIF version |
Description: Obsolete version of rnmptc 7205 as of 17-Apr-2024. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
rnmptcOLD.f | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
rnmptcOLD.b | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) |
rnmptcOLD.a | ⊢ (𝜑 → 𝐴 ≠ ∅) |
Ref | Expression |
---|---|
rnmptcOLD | ⊢ (𝜑 → ran 𝐹 = {𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rnmptcOLD.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
2 | fconstmpt 5737 | . . 3 ⊢ (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
3 | 1, 2 | eqtr4i 2764 | . 2 ⊢ 𝐹 = (𝐴 × {𝐵}) |
4 | rnmptcOLD.b | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) | |
5 | 4, 1 | fmptd 7111 | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) |
6 | 5 | ffnd 6716 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
7 | rnmptcOLD.a | . . 3 ⊢ (𝜑 → 𝐴 ≠ ∅) | |
8 | fconst5 7204 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≠ ∅) → (𝐹 = (𝐴 × {𝐵}) ↔ ran 𝐹 = {𝐵})) | |
9 | 6, 7, 8 | syl2anc 585 | . 2 ⊢ (𝜑 → (𝐹 = (𝐴 × {𝐵}) ↔ ran 𝐹 = {𝐵})) |
10 | 3, 9 | mpbii 232 | 1 ⊢ (𝜑 → ran 𝐹 = {𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∅c0 4322 {csn 4628 ↦ cmpt 5231 × cxp 5674 ran crn 5677 Fn wfn 6536 |
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-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
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-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-fo 6547 df-fv 6549 |
This theorem is referenced by: (None) |
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