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Theorem rnmptssbi 45206
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 5256 . . . . . . 7 𝑥(𝑥𝐴𝐵)
42, 3nfcxfr 2901 . . . . . 6 𝑥𝐹
54nfrn 5966 . . . . 5 𝑥ran 𝐹
6 nfcv 2903 . . . . 5 𝑥𝐶
75, 6nfss 3988 . . . 4 𝑥ran 𝐹𝐶
81, 7nfan 1897 . . 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 5978 . . . 4 (((𝜑 ∧ ran 𝐹𝐶) ∧ 𝑥𝐴) → 𝐵 ∈ ran 𝐹)
149, 13sseldd 3996 . . 3 (((𝜑 ∧ ran 𝐹𝐶) ∧ 𝑥𝐴) → 𝐵𝐶)
158, 14ralrimia 3256 . 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 1537  wnf 1780  wcel 2106  wral 3059  wss 3963  cmpt 5231  ran crn 5690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-fun 6565  df-fn 6566  df-f 6567
This theorem is referenced by:  imassmpt  45208
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