Theorem rnmptcOLD 7065

Description: Obsolete version of rnmptc 7064 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 5640	. . 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 6970	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶)
6	5	ffnd 6585	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴)
7		rnmptcOLD.a	. . 3 ⊢ (𝜑 → 𝐴 ≠ ∅)
8		fconst5 7063	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≠ ∅) → (𝐹 = (𝐴 × {𝐵}) ↔ ran 𝐹 = {𝐵}))
9	6, 7, 8	syl2anc 583	. 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 395   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∅c0 4253  {csn 4558   ↦ cmpt 5153   × cxp 5578  ran crn 5581   Fn wfn 6413

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fo 6424  df-fv 6426

This theorem is referenced by: (None)