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Theorem rnmptssrn 38809
Description: Inclusion relation for two ranges expressed in map-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
rnmptssrn.b ((𝜑𝑥𝐴) → 𝐵𝑉)
rnmptssrn.y ((𝜑𝑥𝐴) → ∃𝑦𝐶 𝐵 = 𝐷)
Assertion
Ref Expression
rnmptssrn (𝜑 → ran (𝑥𝐴𝐵) ⊆ ran (𝑦𝐶𝐷))
Distinct variable groups:   𝑥,𝐴   𝑦,𝐵   𝑥,𝐶   𝑥,𝐷   𝜑,𝑥   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑦)   𝐵(𝑥)   𝐶(𝑦)   𝐷(𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem rnmptssrn
StepHypRef Expression
1 rnmptssrn.y . . . 4 ((𝜑𝑥𝐴) → ∃𝑦𝐶 𝐵 = 𝐷)
2 rnmptssrn.b . . . . 5 ((𝜑𝑥𝐴) → 𝐵𝑉)
3 eqid 2626 . . . . . 6 (𝑦𝐶𝐷) = (𝑦𝐶𝐷)
43elrnmpt 5336 . . . . 5 (𝐵𝑉 → (𝐵 ∈ ran (𝑦𝐶𝐷) ↔ ∃𝑦𝐶 𝐵 = 𝐷))
52, 4syl 17 . . . 4 ((𝜑𝑥𝐴) → (𝐵 ∈ ran (𝑦𝐶𝐷) ↔ ∃𝑦𝐶 𝐵 = 𝐷))
61, 5mpbird 247 . . 3 ((𝜑𝑥𝐴) → 𝐵 ∈ ran (𝑦𝐶𝐷))
76ralrimiva 2965 . 2 (𝜑 → ∀𝑥𝐴 𝐵 ∈ ran (𝑦𝐶𝐷))
8 eqid 2626 . . 3 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
98rnmptss 6348 . 2 (∀𝑥𝐴 𝐵 ∈ ran (𝑦𝐶𝐷) → ran (𝑥𝐴𝐵) ⊆ ran (𝑦𝐶𝐷))
107, 9syl 17 1 (𝜑 → ran (𝑥𝐴𝐵) ⊆ ran (𝑦𝐶𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1992  wral 2912  wrex 2913  wss 3560  cmpt 4678  ran crn 5080
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-fv 5858
This theorem is referenced by:  sge0f1o  39874
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