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Theorem iunrnmptss 32578
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 3071 . . . 4 (∃𝑦 ∈ ran (𝑥𝐴𝐵)𝑧𝐶 ↔ ∃𝑦(𝑦 ∈ ran (𝑥𝐴𝐵) ∧ 𝑧𝐶))
2 iunrnmptss.2 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝐵𝑉)
32ralrimiva 3146 . . . . . . . 8 (𝜑 → ∀𝑥𝐴 𝐵𝑉)
4 eqid 2737 . . . . . . . . 9 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
54elrnmptg 5972 . . . . . . . 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 1921 . . . . 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 2827 . . . . . . . . 9 (𝑦 = 𝐵 → (𝑧𝐶𝑧𝐷))
1211biimpa 476 . . . . . . . 8 ((𝑦 = 𝐵𝑧𝐶) → 𝑧𝐷)
1312reximi 3084 . . . . . . 7 (∃𝑥𝐴 (𝑦 = 𝐵𝑧𝐶) → ∃𝑥𝐴 𝑧𝐷)
149, 13sylbir 235 . . . . . 6 ((∃𝑥𝐴 𝑦 = 𝐵𝑧𝐶) → ∃𝑥𝐴 𝑧𝐷)
1514exlimiv 1930 . . . . 5 (∃𝑦(∃𝑥𝐴 𝑦 = 𝐵𝑧𝐶) → ∃𝑥𝐴 𝑧𝐷)
168, 15biimtrdi 253 . . . 4 (𝜑 → (∃𝑦(𝑦 ∈ ran (𝑥𝐴𝐵) ∧ 𝑧𝐶) → ∃𝑥𝐴 𝑧𝐷))
171, 16biimtrid 242 . . 3 (𝜑 → (∃𝑦 ∈ ran (𝑥𝐴𝐵)𝑧𝐶 → ∃𝑥𝐴 𝑧𝐷))
1817ss2abdv 4066 . 2 (𝜑 → {𝑧 ∣ ∃𝑦 ∈ ran (𝑥𝐴𝐵)𝑧𝐶} ⊆ {𝑧 ∣ ∃𝑥𝐴 𝑧𝐷})
19 df-iun 4993 . 2 𝑦 ∈ ran (𝑥𝐴𝐵)𝐶 = {𝑧 ∣ ∃𝑦 ∈ ran (𝑥𝐴𝐵)𝑧𝐶}
20 df-iun 4993 . 2 𝑥𝐴 𝐷 = {𝑧 ∣ ∃𝑥𝐴 𝑧𝐷}
2118, 19, 203sstr4g 4037 1 (𝜑 𝑦 ∈ ran (𝑥𝐴𝐵)𝐶 𝑥𝐴 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wex 1779  wcel 2108  {cab 2714  wral 3061  wrex 3070  wss 3951   ciun 4991  cmpt 5225  ran crn 5686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-cnv 5693  df-dm 5695  df-rn 5696
This theorem is referenced by:  fnpreimac  32681
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