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Theorem rnmptssbi 39994
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 4899 . . . . . . 7 𝑥(𝑥𝐴𝐵)
42, 3nfcxfr 2900 . . . . . 6 𝑥𝐹
54nfrn 5523 . . . . 5 𝑥ran 𝐹
6 nfcv 2902 . . . . 5 𝑥𝐶
75, 6nfss 3737 . . . 4 𝑥ran 𝐹𝐶
81, 7nfan 1977 . . 3 𝑥(𝜑 ∧ ran 𝐹𝐶)
9 simplr 809 . . . 4 (((𝜑 ∧ ran 𝐹𝐶) ∧ 𝑥𝐴) → ran 𝐹𝐶)
10 simpr 479 . . . . 5 (((𝜑 ∧ ran 𝐹𝐶) ∧ 𝑥𝐴) → 𝑥𝐴)
11 rnmptssbi.3 . . . . . 6 ((𝜑𝑥𝐴) → 𝐵𝑉)
1211adantlr 753 . . . . 5 (((𝜑 ∧ ran 𝐹𝐶) ∧ 𝑥𝐴) → 𝐵𝑉)
132, 10, 12elrnmpt1d 39952 . . . 4 (((𝜑 ∧ ran 𝐹𝐶) ∧ 𝑥𝐴) → 𝐵 ∈ ran 𝐹)
149, 13sseldd 3745 . . 3 (((𝜑 ∧ ran 𝐹𝐶) ∧ 𝑥𝐴) → 𝐵𝐶)
158, 14ralrimia 39832 . 2 ((𝜑 ∧ ran 𝐹𝐶) → ∀𝑥𝐴 𝐵𝐶)
162rnmptss 6556 . . 3 (∀𝑥𝐴 𝐵𝐶 → ran 𝐹𝐶)
1716adantl 473 . 2 ((𝜑 ∧ ∀𝑥𝐴 𝐵𝐶) → ran 𝐹𝐶)
1815, 17impbida 913 1 (𝜑 → (ran 𝐹𝐶 ↔ ∀𝑥𝐴 𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wnf 1857  wcel 2139  wral 3050  wss 3715  cmpt 4881  ran crn 5267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057
This theorem is referenced by:  imassmpt  39998
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