Theorem elrnmpt1s 5918

Description: Elementhood in an image set. (Contributed by Mario Carneiro, 12-Sep-2015.)

Hypotheses

Ref	Expression
rnmpt.1	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
elrnmpt1s.1	⊢ (𝑥 = 𝐷 → 𝐵 = 𝐶)

Assertion

Ref	Expression
elrnmpt1s	⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 ∈ 𝑉) → 𝐶 ∈ ran 𝐹)

Distinct variable groups:   𝑥,𝐶   𝑥,𝐴   𝑥,𝐷

Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem elrnmpt1s

Step	Hyp	Ref	Expression
1		eqid 2737	. . 3 ⊢ 𝐶 = 𝐶
2		elrnmpt1s.1	. . . 4 ⊢ (𝑥 = 𝐷 → 𝐵 = 𝐶)
3	2	rspceeqv 3601	. . 3 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 = 𝐶) → ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)
4	1, 3	mpan2 692	. 2 ⊢ (𝐷 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)
5		rnmpt.1	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
6	5	elrnmpt 5917	. . 3 ⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵))
7	6	biimparc 479	. 2 ⊢ ((∃𝑥 ∈ 𝐴 𝐶 = 𝐵 ∧ 𝐶 ∈ 𝑉) → 𝐶 ∈ ran 𝐹)
8	4, 7	sylan 581	1 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 ∈ 𝑉) → 𝐶 ∈ ran 𝐹)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃wrex 3062   ↦ cmpt 5181  ran crn 5635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5245  ax-pr 5381

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-mpt 5182  df-cnv 5642  df-dm 5644  df-rn 5645

This theorem is referenced by:  wunex2  10663  dfod2  19510  dprd2dlem1  19989  dprd2da  19990  ordtbaslem  23149  subgntr  24068  opnsubg  24069  tgpconncomp  24074  tsmsxplem1  24114  xrge0gsumle  24795  xrge0tsms  24796  minveclem3b  25401  minveclem3  25402  minveclem4  25405  efsubm  26533  dchrisum0fno1  27495  fnpreimac  32766  xrge0tsmsd  33173  esumcvg  34270  esum2d  34277  msubco  35753  suprubrnmpt2  45639  infxrlbrnmpt2  45797  sge0xaddlem1  46820