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Theorem iunrnmptss 31797
Description: A subset relation for an indexed union over the range of function expressed as a mapping. (Contributed by Thierry Arnoux, 27-Mar-2018.)
Hypotheses
Ref Expression
iunrnmptss.1 (𝑦 = 𝐵𝐶 = 𝐷)
iunrnmptss.2 ((𝜑𝑥𝐴) → 𝐵𝑉)
Assertion
Ref Expression
iunrnmptss (𝜑 𝑦 ∈ ran (𝑥𝐴𝐵)𝐶 𝑥𝐴 𝐷)
Distinct variable groups:   𝑦,𝐴   𝑥,𝐶   𝑦,𝐷   𝜑,𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥,𝑦)   𝐶(𝑦)   𝐷(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem iunrnmptss
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-rex 3072 . . . 4 (∃𝑦 ∈ ran (𝑥𝐴𝐵)𝑧𝐶 ↔ ∃𝑦(𝑦 ∈ ran (𝑥𝐴𝐵) ∧ 𝑧𝐶))
2 iunrnmptss.2 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝐵𝑉)
32ralrimiva 3147 . . . . . . . 8 (𝜑 → ∀𝑥𝐴 𝐵𝑉)
4 eqid 2733 . . . . . . . . 9 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
54elrnmptg 5959 . . . . . . . 8 (∀𝑥𝐴 𝐵𝑉 → (𝑦 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑦 = 𝐵))
63, 5syl 17 . . . . . . 7 (𝜑 → (𝑦 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑦 = 𝐵))
76anbi1d 631 . . . . . 6 (𝜑 → ((𝑦 ∈ ran (𝑥𝐴𝐵) ∧ 𝑧𝐶) ↔ (∃𝑥𝐴 𝑦 = 𝐵𝑧𝐶)))
87exbidv 1925 . . . . 5 (𝜑 → (∃𝑦(𝑦 ∈ ran (𝑥𝐴𝐵) ∧ 𝑧𝐶) ↔ ∃𝑦(∃𝑥𝐴 𝑦 = 𝐵𝑧𝐶)))
9 r19.41v 3189 . . . . . . 7 (∃𝑥𝐴 (𝑦 = 𝐵𝑧𝐶) ↔ (∃𝑥𝐴 𝑦 = 𝐵𝑧𝐶))
10 iunrnmptss.1 . . . . . . . . . 10 (𝑦 = 𝐵𝐶 = 𝐷)
1110eleq2d 2820 . . . . . . . . 9 (𝑦 = 𝐵 → (𝑧𝐶𝑧𝐷))
1211biimpa 478 . . . . . . . 8 ((𝑦 = 𝐵𝑧𝐶) → 𝑧𝐷)
1312reximi 3085 . . . . . . 7 (∃𝑥𝐴 (𝑦 = 𝐵𝑧𝐶) → ∃𝑥𝐴 𝑧𝐷)
149, 13sylbir 234 . . . . . 6 ((∃𝑥𝐴 𝑦 = 𝐵𝑧𝐶) → ∃𝑥𝐴 𝑧𝐷)
1514exlimiv 1934 . . . . 5 (∃𝑦(∃𝑥𝐴 𝑦 = 𝐵𝑧𝐶) → ∃𝑥𝐴 𝑧𝐷)
168, 15syl6bi 253 . . . 4 (𝜑 → (∃𝑦(𝑦 ∈ ran (𝑥𝐴𝐵) ∧ 𝑧𝐶) → ∃𝑥𝐴 𝑧𝐷))
171, 16biimtrid 241 . . 3 (𝜑 → (∃𝑦 ∈ ran (𝑥𝐴𝐵)𝑧𝐶 → ∃𝑥𝐴 𝑧𝐷))
1817ss2abdv 4061 . 2 (𝜑 → {𝑧 ∣ ∃𝑦 ∈ ran (𝑥𝐴𝐵)𝑧𝐶} ⊆ {𝑧 ∣ ∃𝑥𝐴 𝑧𝐷})
19 df-iun 5000 . 2 𝑦 ∈ ran (𝑥𝐴𝐵)𝐶 = {𝑧 ∣ ∃𝑦 ∈ ran (𝑥𝐴𝐵)𝑧𝐶}
20 df-iun 5000 . 2 𝑥𝐴 𝐷 = {𝑧 ∣ ∃𝑥𝐴 𝑧𝐷}
2118, 19, 203sstr4g 4028 1 (𝜑 𝑦 ∈ ran (𝑥𝐴𝐵)𝐶 𝑥𝐴 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wex 1782  wcel 2107  {cab 2710  wral 3062  wrex 3071  wss 3949   ciun 4998  cmpt 5232  ran crn 5678
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-cnv 5685  df-dm 5687  df-rn 5688
This theorem is referenced by:  fnpreimac  31896
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