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Theorem rnmptssbi 45267
Description: The range of a function given by the maps-to notation as a subset. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
Hypotheses
Ref Expression
rnmptssbi.1 𝑥𝜑
rnmptssbi.2 𝐹 = (𝑥𝐴𝐵)
rnmptssbi.3 ((𝜑𝑥𝐴) → 𝐵𝑉)
Assertion
Ref Expression
rnmptssbi (𝜑 → (ran 𝐹𝐶 ↔ ∀𝑥𝐴 𝐵𝐶))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem rnmptssbi
StepHypRef Expression
1 rnmptssbi.1 . . . 4 𝑥𝜑
2 rnmptssbi.2 . . . . . . 7 𝐹 = (𝑥𝐴𝐵)
3 nfmpt1 5250 . . . . . . 7 𝑥(𝑥𝐴𝐵)
42, 3nfcxfr 2903 . . . . . 6 𝑥𝐹
54nfrn 5963 . . . . 5 𝑥ran 𝐹
6 nfcv 2905 . . . . 5 𝑥𝐶
75, 6nfss 3976 . . . 4 𝑥ran 𝐹𝐶
81, 7nfan 1899 . . 3 𝑥(𝜑 ∧ ran 𝐹𝐶)
9 simplr 769 . . . 4 (((𝜑 ∧ ran 𝐹𝐶) ∧ 𝑥𝐴) → ran 𝐹𝐶)
10 simpr 484 . . . . 5 (((𝜑 ∧ ran 𝐹𝐶) ∧ 𝑥𝐴) → 𝑥𝐴)
11 rnmptssbi.3 . . . . . 6 ((𝜑𝑥𝐴) → 𝐵𝑉)
1211adantlr 715 . . . . 5 (((𝜑 ∧ ran 𝐹𝐶) ∧ 𝑥𝐴) → 𝐵𝑉)
132, 10, 12elrnmpt1d 5975 . . . 4 (((𝜑 ∧ ran 𝐹𝐶) ∧ 𝑥𝐴) → 𝐵 ∈ ran 𝐹)
149, 13sseldd 3984 . . 3 (((𝜑 ∧ ran 𝐹𝐶) ∧ 𝑥𝐴) → 𝐵𝐶)
158, 14ralrimia 3258 . 2 ((𝜑 ∧ ran 𝐹𝐶) → ∀𝑥𝐴 𝐵𝐶)
162rnmptss 7143 . . 3 (∀𝑥𝐴 𝐵𝐶 → ran 𝐹𝐶)
1716adantl 481 . 2 ((𝜑 ∧ ∀𝑥𝐴 𝐵𝐶) → ran 𝐹𝐶)
1815, 17impbida 801 1 (𝜑 → (ran 𝐹𝐶 ↔ ∀𝑥𝐴 𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wnf 1783  wcel 2108  wral 3061  wss 3951  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-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-fun 6563  df-fn 6564  df-f 6565
This theorem is referenced by:  imassmpt  45269
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