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Mirrors > Home > MPE Home > Th. List > rnmptcOLD | Structured version Visualization version GIF version |
Description: Obsolete version of rnmptc 6962 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 5607 | . . 3 ⊢ (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
3 | 1, 2 | eqtr4i 2846 | . 2 ⊢ 𝐹 = (𝐴 × {𝐵}) |
4 | rnmptcOLD.b | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) | |
5 | 4, 1 | fmptd 6871 | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) |
6 | 5 | ffnd 6508 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
7 | rnmptcOLD.a | . . 3 ⊢ (𝜑 → 𝐴 ≠ ∅) | |
8 | fconst5 6961 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≠ ∅) → (𝐹 = (𝐴 × {𝐵}) ↔ ran 𝐹 = {𝐵})) | |
9 | 6, 7, 8 | syl2anc 586 | . 2 ⊢ (𝜑 → (𝐹 = (𝐴 × {𝐵}) ↔ ran 𝐹 = {𝐵})) |
10 | 3, 9 | mpbii 235 | 1 ⊢ (𝜑 → ran 𝐹 = {𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ≠ wne 3015 ∅c0 4284 {csn 4560 ↦ cmpt 5139 × cxp 5546 ran crn 5549 Fn wfn 6343 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fo 6354 df-fv 6356 |
This theorem is referenced by: (None) |
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