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Theorem rnmptbddlem 40368
Description: Boundness of the range of a function in maps-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 3401 . . . . . . . . 9 𝑧 ∈ V
3 eqid 2778 . . . . . . . . . 10 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
43elrnmpt 5618 . . . . . . . . 9 (𝑧 ∈ V → (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵))
52, 4ax-mp 5 . . . . . . . 8 (𝑧 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 𝑧 = 𝐵)
65biimpi 208 . . . . . . 7 (𝑧 ∈ ran (𝑥𝐴𝐵) → ∃𝑥𝐴 𝑧 = 𝐵)
76adantl 475 . . . . . 6 ((((𝜑𝑦 ∈ ℝ) ∧ ∀𝑥𝐴 𝐵𝑦) ∧ 𝑧 ∈ ran (𝑥𝐴𝐵)) → ∃𝑥𝐴 𝑧 = 𝐵)
8 rnmptbddlem.x . . . . . . . . . 10 𝑥𝜑
9 nfv 1957 . . . . . . . . . 10 𝑥 𝑦 ∈ ℝ
108, 9nfan 1946 . . . . . . . . 9 𝑥(𝜑𝑦 ∈ ℝ)
11 nfra1 3123 . . . . . . . . 9 𝑥𝑥𝐴 𝐵𝑦
1210, 11nfan 1946 . . . . . . . 8 𝑥((𝜑𝑦 ∈ ℝ) ∧ ∀𝑥𝐴 𝐵𝑦)
13 nfv 1957 . . . . . . . 8 𝑥 𝑧𝑦
14 rspa 3112 . . . . . . . . . . . 12 ((∀𝑥𝐴 𝐵𝑦𝑥𝐴) → 𝐵𝑦)
15143adant3 1123 . . . . . . . . . . 11 ((∀𝑥𝐴 𝐵𝑦𝑥𝐴𝑧 = 𝐵) → 𝐵𝑦)
16 simp3 1129 . . . . . . . . . . 11 ((∀𝑥𝐴 𝐵𝑦𝑥𝐴𝑧 = 𝐵) → 𝑧 = 𝐵)
17 simpr 479 . . . . . . . . . . . 12 ((𝐵𝑦𝑧 = 𝐵) → 𝑧 = 𝐵)
18 simpl 476 . . . . . . . . . . . 12 ((𝐵𝑦𝑧 = 𝐵) → 𝐵𝑦)
1917, 18eqbrtrd 4908 . . . . . . . . . . 11 ((𝐵𝑦𝑧 = 𝐵) → 𝑧𝑦)
2015, 16, 19syl2anc 579 . . . . . . . . . 10 ((∀𝑥𝐴 𝐵𝑦𝑥𝐴𝑧 = 𝐵) → 𝑧𝑦)
21203exp 1109 . . . . . . . . 9 (∀𝑥𝐴 𝐵𝑦 → (𝑥𝐴 → (𝑧 = 𝐵𝑧𝑦)))
2221adantl 475 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑥𝐴 𝐵𝑦) → (𝑥𝐴 → (𝑧 = 𝐵𝑧𝑦)))
2312, 13, 22rexlimd 3208 . . . . . . 7 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑥𝐴 𝐵𝑦) → (∃𝑥𝐴 𝑧 = 𝐵𝑧𝑦))
2423imp 397 . . . . . 6 ((((𝜑𝑦 ∈ ℝ) ∧ ∀𝑥𝐴 𝐵𝑦) ∧ ∃𝑥𝐴 𝑧 = 𝐵) → 𝑧𝑦)
257, 24syldan 585 . . . . 5 ((((𝜑𝑦 ∈ ℝ) ∧ ∀𝑥𝐴 𝐵𝑦) ∧ 𝑧 ∈ ran (𝑥𝐴𝐵)) → 𝑧𝑦)
2625ralrimiva 3148 . . . 4 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑥𝐴 𝐵𝑦) → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
2726ex 403 . . 3 ((𝜑𝑦 ∈ ℝ) → (∀𝑥𝐴 𝐵𝑦 → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦))
2827reximdva 3198 . 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 198  wa 386  w3a 1071   = wceq 1601  wnf 1827  wcel 2107  wral 3090  wrex 3091  Vcvv 3398   class class class wbr 4886  cmpt 4965  ran crn 5356  cr 10271  cle 10412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4887  df-opab 4949  df-mpt 4966  df-cnv 5363  df-dm 5365  df-rn 5366
This theorem is referenced by:  rnmptbdd  40369
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