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Theorem iunrnmptss 30332
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 3115 . . . 4 (∃𝑦 ∈ ran (𝑥𝐴𝐵)𝑧𝐶 ↔ ∃𝑦(𝑦 ∈ ran (𝑥𝐴𝐵) ∧ 𝑧𝐶))
2 iunrnmptss.2 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝐵𝑉)
32ralrimiva 3152 . . . . . . . 8 (𝜑 → ∀𝑥𝐴 𝐵𝑉)
4 eqid 2801 . . . . . . . . 9 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
54elrnmptg 5799 . . . . . . . 8 (∀𝑥𝐴 𝐵𝑉 → (𝑦 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑦 = 𝐵))
63, 5syl 17 . . . . . . 7 (𝜑 → (𝑦 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑦 = 𝐵))
76anbi1d 632 . . . . . 6 (𝜑 → ((𝑦 ∈ ran (𝑥𝐴𝐵) ∧ 𝑧𝐶) ↔ (∃𝑥𝐴 𝑦 = 𝐵𝑧𝐶)))
87exbidv 1922 . . . . 5 (𝜑 → (∃𝑦(𝑦 ∈ ran (𝑥𝐴𝐵) ∧ 𝑧𝐶) ↔ ∃𝑦(∃𝑥𝐴 𝑦 = 𝐵𝑧𝐶)))
9 r19.41v 3303 . . . . . . 7 (∃𝑥𝐴 (𝑦 = 𝐵𝑧𝐶) ↔ (∃𝑥𝐴 𝑦 = 𝐵𝑧𝐶))
10 iunrnmptss.1 . . . . . . . . . 10 (𝑦 = 𝐵𝐶 = 𝐷)
1110eleq2d 2878 . . . . . . . . 9 (𝑦 = 𝐵 → (𝑧𝐶𝑧𝐷))
1211biimpa 480 . . . . . . . 8 ((𝑦 = 𝐵𝑧𝐶) → 𝑧𝐷)
1312reximi 3209 . . . . . . 7 (∃𝑥𝐴 (𝑦 = 𝐵𝑧𝐶) → ∃𝑥𝐴 𝑧𝐷)
149, 13sylbir 238 . . . . . 6 ((∃𝑥𝐴 𝑦 = 𝐵𝑧𝐶) → ∃𝑥𝐴 𝑧𝐷)
1514exlimiv 1931 . . . . 5 (∃𝑦(∃𝑥𝐴 𝑦 = 𝐵𝑧𝐶) → ∃𝑥𝐴 𝑧𝐷)
168, 15syl6bi 256 . . . 4 (𝜑 → (∃𝑦(𝑦 ∈ ran (𝑥𝐴𝐵) ∧ 𝑧𝐶) → ∃𝑥𝐴 𝑧𝐷))
171, 16syl5bi 245 . . 3 (𝜑 → (∃𝑦 ∈ ran (𝑥𝐴𝐵)𝑧𝐶 → ∃𝑥𝐴 𝑧𝐷))
1817ss2abdv 3994 . 2 (𝜑 → {𝑧 ∣ ∃𝑦 ∈ ran (𝑥𝐴𝐵)𝑧𝐶} ⊆ {𝑧 ∣ ∃𝑥𝐴 𝑧𝐷})
19 df-iun 4886 . 2 𝑦 ∈ ran (𝑥𝐴𝐵)𝐶 = {𝑧 ∣ ∃𝑦 ∈ ran (𝑥𝐴𝐵)𝑧𝐶}
20 df-iun 4886 . 2 𝑥𝐴 𝐷 = {𝑧 ∣ ∃𝑥𝐴 𝑧𝐷}
2118, 19, 203sstr4g 3963 1 (𝜑 𝑦 ∈ ran (𝑥𝐴𝐵)𝐶 𝑥𝐴 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wex 1781  wcel 2112  {cab 2779  wral 3109  wrex 3110  wss 3884   ciun 4884  cmpt 5113  ran crn 5524
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
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 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-cnv 5531  df-dm 5533  df-rn 5534
This theorem is referenced by:  fnpreimac  30437
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