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Theorem rnmptssbi 39791
Description: The range of an operation 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 4780 . . . . . . 7 𝑥(𝑥𝐴𝐵)
42, 3nfcxfr 2791 . . . . . 6 𝑥𝐹
54nfrn 5400 . . . . 5 𝑥ran 𝐹
6 nfcv 2793 . . . . 5 𝑥𝐶
75, 6nfss 3629 . . . 4 𝑥ran 𝐹𝐶
81, 7nfan 1868 . . 3 𝑥(𝜑 ∧ ran 𝐹𝐶)
9 simplr 807 . . . 4 (((𝜑 ∧ ran 𝐹𝐶) ∧ 𝑥𝐴) → ran 𝐹𝐶)
10 simpr 476 . . . . 5 (((𝜑 ∧ ran 𝐹𝐶) ∧ 𝑥𝐴) → 𝑥𝐴)
11 rnmptssbi.3 . . . . . 6 ((𝜑𝑥𝐴) → 𝐵𝑉)
1211adantlr 751 . . . . 5 (((𝜑 ∧ ran 𝐹𝐶) ∧ 𝑥𝐴) → 𝐵𝑉)
132, 10, 12elrnmpt1d 39749 . . . 4 (((𝜑 ∧ ran 𝐹𝐶) ∧ 𝑥𝐴) → 𝐵 ∈ ran 𝐹)
149, 13sseldd 3637 . . 3 (((𝜑 ∧ ran 𝐹𝐶) ∧ 𝑥𝐴) → 𝐵𝐶)
158, 14ralrimia 39629 . 2 ((𝜑 ∧ ran 𝐹𝐶) → ∀𝑥𝐴 𝐵𝐶)
162rnmptss 6432 . . 3 (∀𝑥𝐴 𝐵𝐶 → ran 𝐹𝐶)
1716adantl 481 . 2 ((𝜑 ∧ ∀𝑥𝐴 𝐵𝐶) → ran 𝐹𝐶)
1815, 17impbida 895 1 (𝜑 → (ran 𝐹𝐶 ↔ ∀𝑥𝐴 𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wnf 1748  wcel 2030  wral 2941  wss 3607  cmpt 4762  ran crn 5144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934
This theorem is referenced by:  imassmpt  39795
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