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Theorem rnmptbddlem 39972
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 3343 . . . . . . . . 9 𝑧 ∈ V
3 eqid 2760 . . . . . . . . . 10 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
43elrnmpt 5527 . . . . . . . . 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 473 . . . . . 6 ((((𝜑𝑦 ∈ ℝ) ∧ ∀𝑥𝐴 𝐵𝑦) ∧ 𝑧 ∈ ran (𝑥𝐴𝐵)) → ∃𝑥𝐴 𝑧 = 𝐵)
8 rnmptbddlem.x . . . . . . . . . 10 𝑥𝜑
9 nfv 1992 . . . . . . . . . 10 𝑥 𝑦 ∈ ℝ
108, 9nfan 1977 . . . . . . . . 9 𝑥(𝜑𝑦 ∈ ℝ)
11 nfra1 3079 . . . . . . . . 9 𝑥𝑥𝐴 𝐵𝑦
1210, 11nfan 1977 . . . . . . . 8 𝑥((𝜑𝑦 ∈ ℝ) ∧ ∀𝑥𝐴 𝐵𝑦)
13 nfv 1992 . . . . . . . 8 𝑥 𝑧𝑦
14 rspa 3068 . . . . . . . . . . . 12 ((∀𝑥𝐴 𝐵𝑦𝑥𝐴) → 𝐵𝑦)
15143adant3 1127 . . . . . . . . . . 11 ((∀𝑥𝐴 𝐵𝑦𝑥𝐴𝑧 = 𝐵) → 𝐵𝑦)
16 simp3 1133 . . . . . . . . . . 11 ((∀𝑥𝐴 𝐵𝑦𝑥𝐴𝑧 = 𝐵) → 𝑧 = 𝐵)
17 simpr 479 . . . . . . . . . . . 12 ((𝐵𝑦𝑧 = 𝐵) → 𝑧 = 𝐵)
18 simpl 474 . . . . . . . . . . . 12 ((𝐵𝑦𝑧 = 𝐵) → 𝐵𝑦)
1917, 18eqbrtrd 4826 . . . . . . . . . . 11 ((𝐵𝑦𝑧 = 𝐵) → 𝑧𝑦)
2015, 16, 19syl2anc 696 . . . . . . . . . 10 ((∀𝑥𝐴 𝐵𝑦𝑥𝐴𝑧 = 𝐵) → 𝑧𝑦)
21203exp 1113 . . . . . . . . 9 (∀𝑥𝐴 𝐵𝑦 → (𝑥𝐴 → (𝑧 = 𝐵𝑧𝑦)))
2221adantl 473 . . . . . . . 8 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑥𝐴 𝐵𝑦) → (𝑥𝐴 → (𝑧 = 𝐵𝑧𝑦)))
2312, 13, 22rexlimd 3164 . . . . . . 7 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑥𝐴 𝐵𝑦) → (∃𝑥𝐴 𝑧 = 𝐵𝑧𝑦))
2423imp 444 . . . . . 6 ((((𝜑𝑦 ∈ ℝ) ∧ ∀𝑥𝐴 𝐵𝑦) ∧ ∃𝑥𝐴 𝑧 = 𝐵) → 𝑧𝑦)
257, 24syldan 488 . . . . 5 ((((𝜑𝑦 ∈ ℝ) ∧ ∀𝑥𝐴 𝐵𝑦) ∧ 𝑧 ∈ ran (𝑥𝐴𝐵)) → 𝑧𝑦)
2625ralrimiva 3104 . . . 4 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑥𝐴 𝐵𝑦) → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
2726ex 449 . . 3 ((𝜑𝑦 ∈ ℝ) → (∀𝑥𝐴 𝐵𝑦 → ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦))
2827reximdva 3155 . 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 1072   = wceq 1632  wnf 1857  wcel 2139  wral 3050  wrex 3051  Vcvv 3340   class class class wbr 4804  cmpt 4881  ran crn 5267  cr 10147  cle 10287
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-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  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-br 4805  df-opab 4865  df-mpt 4882  df-cnv 5274  df-dm 5276  df-rn 5277
This theorem is referenced by:  rnmptbdd  39973
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