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Theorem rnmptssrn 41673
 Description: Inclusion relation for two ranges expressed in maps-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 eqid 2824 . . . 4 (𝑦𝐶𝐷) = (𝑦𝐶𝐷)
2 rnmptssrn.y . . . 4 ((𝜑𝑥𝐴) → ∃𝑦𝐶 𝐵 = 𝐷)
3 rnmptssrn.b . . . 4 ((𝜑𝑥𝐴) → 𝐵𝑉)
41, 2, 3elrnmptd 41671 . . 3 ((𝜑𝑥𝐴) → 𝐵 ∈ ran (𝑦𝐶𝐷))
54ralrimiva 3177 . 2 (𝜑 → ∀𝑥𝐴 𝐵 ∈ ran (𝑦𝐶𝐷))
6 eqid 2824 . . 3 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
76rnmptss 6875 . 2 (∀𝑥𝐴 𝐵 ∈ ran (𝑦𝐶𝐷) → ran (𝑥𝐴𝐵) ⊆ ran (𝑦𝐶𝐷))
85, 7syl 17 1 (𝜑 → ran (𝑥𝐴𝐵) ⊆ ran (𝑦𝐶𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2115  ∀wral 3133  ∃wrex 3134   ⊆ wss 3919   ↦ cmpt 5133  ran crn 5544 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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-fv 6352 This theorem is referenced by:  sge0f1o  42887
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