Theorem rnmptcOLD 7083

Description: Obsolete version of rnmptc 7082 as of 17-Apr-2024. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
rnmptcOLD.f	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
rnmptcOLD.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)
rnmptcOLD.a	⊢ (𝜑 → 𝐴 ≠ ∅)

Assertion

Ref	Expression
rnmptcOLD	⊢ (𝜑 → ran 𝐹 = {𝐵})

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝜑,𝑥

Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem rnmptcOLD

Step	Hyp	Ref	Expression
1		rnmptcOLD.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
2		fconstmpt 5649	. . 3 ⊢ (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵)
3	1, 2	eqtr4i 2769	. 2 ⊢ 𝐹 = (𝐴 × {𝐵})
4		rnmptcOLD.b	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)
5	4, 1	fmptd 6988	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶)
6	5	ffnd 6601	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴)
7		rnmptcOLD.a	. . 3 ⊢ (𝜑 → 𝐴 ≠ ∅)
8		fconst5 7081	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≠ ∅) → (𝐹 = (𝐴 × {𝐵}) ↔ ran 𝐹 = {𝐵}))
9	6, 7, 8	syl2anc 584	. 2 ⊢ (𝜑 → (𝐹 = (𝐴 × {𝐵}) ↔ ran 𝐹 = {𝐵}))
10	3, 9	mpbii 232	1 ⊢ (𝜑 → ran 𝐹 = {𝐵})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∅c0 4256  {csn 4561   ↦ cmpt 5157   × cxp 5587  ran crn 5590   Fn wfn 6428

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fo 6439  df-fv 6441

This theorem is referenced by: (None)