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Theorem rnmptbddlem 39769
Description: Boundness of the range of a function in map-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
rnmptbddlem.x 𝑥𝜑
rnmptbddlem.b (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)
Assertion
Ref Expression
rnmptbddlem (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
Distinct variable groups:   𝑧,𝐴   𝑧,𝐵   𝜑,𝑦,𝑧   𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem rnmptbddlem
StepHypRef Expression
1 rnmptbddlem.b . 2 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)
2 vex 3234 . . . . . . . . 9 𝑧 ∈ V
3 eqid 2651 . . . . . . . . . 10 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
43elrnmpt 5404 . . . . . . . . 9 (𝑧 ∈ V → (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵))
52, 4ax-mp 5 . . . . . . . 8 (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵)
65biimpi 206 . . . . . . 7 (𝑧 ∈ ran (𝑥𝐴𝐵) → ∃𝑥𝐴 𝑧 = 𝐵)
76adantl 481 . . . . . 6 ((((𝜑𝑦 ∈ ℝ) ∧ ∀𝑥𝐴 𝐵𝑦) ∧ 𝑧 ∈ ran (𝑥𝐴𝐵)) → ∃𝑥𝐴 𝑧 = 𝐵)
8 rnmptbddlem.x . . . . . . . . . 10 𝑥𝜑
9 nfv 1883 . . . . . . . . . 10 𝑥 𝑦 ∈ ℝ
108, 9nfan 1868 . . . . . . . . 9 𝑥(𝜑𝑦 ∈ ℝ)
11 nfra1 2970 . . . . . . . . 9 𝑥𝑥𝐴 𝐵𝑦
1210, 11nfan 1868 . . . . . . . 8 𝑥((𝜑𝑦 ∈ ℝ) ∧ ∀𝑥𝐴 𝐵𝑦)
13 nfv 1883 . . . . . . . 8 𝑥 𝑧𝑦
14 rspa 2959 . . . . . . . . . . . 12 ((∀𝑥𝐴 𝐵𝑦𝑥𝐴) → 𝐵𝑦)
15143adant3 1101 . . . . . . . . . . 11 ((∀𝑥𝐴 𝐵𝑦𝑥𝐴𝑧 = 𝐵) → 𝐵𝑦)
16 simp3 1083 . . . . . . . . . . 11 ((∀𝑥𝐴 𝐵𝑦𝑥𝐴𝑧 = 𝐵) → 𝑧 = 𝐵)
17 simpr 476 . . . . . . . . . . . 12 ((𝐵𝑦𝑧 = 𝐵) → 𝑧 = 𝐵)
18 simpl 472 . . . . . . . . . . . 12 ((𝐵𝑦𝑧 = 𝐵) → 𝐵𝑦)
1917, 18eqbrtrd 4707 . . . . . . . . . . 11 ((𝐵𝑦𝑧 = 𝐵) → 𝑧𝑦)
2015, 16, 19syl2anc 694 . . . . . . . . . 10 ((∀𝑥𝐴 𝐵𝑦𝑥𝐴𝑧 = 𝐵) → 𝑧𝑦)
21203exp 1283 . . . . . . . . 9 (∀𝑥𝐴 𝐵𝑦 → (𝑥𝐴 → (𝑧 = 𝐵𝑧𝑦)))
2221adantl 481 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑥𝐴 𝐵𝑦) → (𝑥𝐴 → (𝑧 = 𝐵𝑧𝑦)))
2312, 13, 22rexlimd 3055 . . . . . . 7 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑥𝐴 𝐵𝑦) → (∃𝑥𝐴 𝑧 = 𝐵𝑧𝑦))
2423imp 444 . . . . . 6 ((((𝜑𝑦 ∈ ℝ) ∧ ∀𝑥𝐴 𝐵𝑦) ∧ ∃𝑥𝐴 𝑧 = 𝐵) → 𝑧𝑦)
257, 24syldan 486 . . . . 5 ((((𝜑𝑦 ∈ ℝ) ∧ ∀𝑥𝐴 𝐵𝑦) ∧ 𝑧 ∈ ran (𝑥𝐴𝐵)) → 𝑧𝑦)
2625ralrimiva 2995 . . . 4 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑥𝐴 𝐵𝑦) → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
2726ex 449 . . 3 ((𝜑𝑦 ∈ ℝ) → (∀𝑥𝐴 𝐵𝑦 → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦))
2827reximdva 3046 . 2 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦))
291, 28mpd 15 1 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wnf 1748  wcel 2030  wral 2941  wrex 2942  Vcvv 3231   class class class wbr 4685  cmpt 4762  ran crn 5144  cr 9973  cle 10113
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-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  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-br 4686  df-opab 4746  df-mpt 4763  df-cnv 5151  df-dm 5153  df-rn 5154
This theorem is referenced by:  rnmptbdd  39770
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